Signature codes for weighted binary adder channel and multimedia fingerprinting

TL;DR

This paper investigates the theoretical limits and constructions of signature codes for weighted binary adder channels and multimedia fingerprinting, providing bounds, exact values, and new code designs with applications to noisy environments.

Contribution

It derives tight bounds, determines exact code sizes for specific cases, and introduces new signature code constructions with efficient decoding for multimedia fingerprinting.

Findings

Upper bounds relate signature codes to bipartite graphs and $B_t$ codes.

Exact values of $A(n,2,2)$ and $A(n,3,2)$ are established for many $n$.

New code constructions enable efficient decoding and are applicable to noisy scenarios.

Abstract

In this paper, we study the signature codes for weighted binary adder channel (WbAC) and collusion-resistant multimedia fingerprinting. Let $A(n,t)$ denote the maximum cardinality of a $t$-signature code of length $n$, and $A(n,w,t)$ denote the maximum cardinality of a $t$-signature code of length $n$ and constant weight $w$. First, we derive asymptotic and general upper bounds of $A(n,t)$ by relating signature codes to $B_t$ codes and bipartite graphs with large girth respectively, and also show the upper bounds are tight for certain cases. Second, we determine the exact values of $A(n,2,2)$ and $A(n,3,2)$ for infinitely many $n$ by connecting signature codes with $C_4$-free graphs and union-free families, respectively. Third, we provide two explicit constructions for $t$-signature codes which have efficient decoding algorithms and applications to two-level signature codes. Furthermore, we show from the geometric viewpoint that there does not exist any binary code with complete traceability for noisy WbAC and multimedia fingerprinting. A new type of signature codes with a weaker requirement than complete traceability is introduced for the noisy scenario.

Tables

Table 1. TABLE I : The values of A ​ ( n , 2 , 2 ) 𝐴 𝑛 2 2 A(n,2,2) for 2 ≤ n ≤ 21 2 𝑛 21 2\leq n\leq 21  ( [ 13 ] )

$n$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$
$A(n,2,2)$	$1$	$3$	$4$	$6$	$7$	$9$	$11$	$13$	$16$	$18$
$n$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$	$20$	$21$
$A(n,2,2)$	$21$	$24$	$27$	$30$	$33$	$36$	$39$	$42$	$46$	$50$

Equations

$$\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}_{j}=\sum\limits_{i=1}^{n}c_{j}(i)\mathchoice{\mbox{\boldmath$\displaystyle f$}}{\mbox{\boldmath$\textstyle f$}}{\mbox{\boldmath$\scriptstyle f$}}{\mbox{\boldmath$\scriptscriptstyle f$}}_{i},$$

$$\mathchoice{\mbox{\boldmath$\displaystyle y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle y$}}{\mbox{\boldmath$\scriptscriptstyle y$}}_{j}=\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}+\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}_{j}$$

$$\hat{\mathchoice{\mbox{\boldmath$\displaystyle y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle y$}}{\mbox{\boldmath$\scriptscriptstyle y$}}}=\sum\limits_{j\in J}\lambda_{j}\mathchoice{\mbox{\boldmath$\displaystyle y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle y$}}{\mbox{\boldmath$\scriptscriptstyle y$}}_{j}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=\sum_{j\in J}\lambda_{j}(\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}+\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}_{j}),}$$

$$\hat{\mathchoice{\mbox{\boldmath$\displaystyle y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle y$}}{\mbox{\boldmath$\scriptscriptstyle y$}}}=\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle x$}}+\sum\limits_{j\in J}\lambda_{j}\mathchoice{\mbox{\boldmath$\displaystyle w$}}{\mbox{\boldmath$\textstyle w$}}{\mbox{\boldmath$\scriptstyle w$}}{\mbox{\boldmath$\scriptscriptstyle w$}}_{j},$$

$$r(k)=\langle\hat{\mathchoice{\mbox{\boldmath$\displaystyle y$}}{\mbox{\boldmath$\textstyle y$}}{\mbox{\boldmath$\scriptstyle y$}}{\mbox{\boldmath$\scriptscriptstyle y$}}}-\mathchoice{\mbox{\boldmath$\displaystyle x$}}{\mbox{\boldmath$\textstyle x$}}{\mbox{\boldmath$\scriptstyle x$}}{\mbox{\boldmath$\scriptscriptstyle x$}},\mathchoice{\mbox{\boldmath$\displaystyle f$}}{\mbox{\boldmath$\textstyle f$}}{\mbox{\boldmath$\scriptstyle f$}}{\mbox{\boldmath$\scriptscriptstyle f$}}_{k}\rangle=\Big{\langle}\sum\limits_{i=1}^{n}\sum\limits_{j\in J}\lambda_{j}c_{j}(i)\mathchoice{\mbox{\boldmath$\displaystyle f$}}{\mbox{\boldmath$\textstyle f$}}{\mbox{\boldmath$\scriptstyle f$}}{\mbox{\boldmath$\scriptscriptstyle f$}}_{i},\mathchoice{\mbox{\boldmath$\displaystyle f$}}{\mbox{\boldmath$\textstyle f$}}{\mbox{\boldmath$\scriptstyle f$}}{\mbox{\boldmath$\scriptscriptstyle f$}}_{k}\Big{\rangle}=\sum\limits_{j\in J}\lambda_{j}c_{j}(k)$$

$$\mathchoice{\mbox{\boldmath$\displaystyle r$}}{\mbox{\boldmath$\textstyle r$}}{\mbox{\boldmath$\scriptstyle r$}}{\mbox{\boldmath$\scriptscriptstyle r$}}=(r(1),\ldots,r(n))=\sum_{j\in J}\lambda_{j}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j}.$$

$$\sigma(r(k))=\left\{\begin{array}[]{ll}\{r(k)\},&r(k)=0\ \text{or}\ 1,\\ \{0,1\},&0<r(k)<1,\end{array}\right.$$

$$\sum_{\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j}\neq\sum_{\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{k}$$

$$\sum_{h=1}^{s}\lambda_{j_{h}}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j_{h}}\neq\sum_{l=1}^{m}\lambda_{k_{l}}^{\prime}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{k_{l}}$$

$$\sum_{h=1}^{s}\lambda_{j_{h}}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j_{h}}=\sum_{l=1}^{m}\lambda_{k_{l}}^{\prime}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{k_{l}}.$$

$$\sum_{\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j}=\sum_{\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{k}$$

$$\mathcal{P}({\cal C})=\Big{\{}\sum\limits_{i=1}^{m}\lambda_{i}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{i}:\ \sum\limits_{i=1}^{m}\lambda_{i}=1\ \text{and}\ \lambda_{i}>0,\ \forall 1\leq i\leq m\Big{\}}.$$

$$\mathcal{P}({\mathcal{C}}_1)\cap \mathcal{P}({\mathcal{C}}_2)=\varnothing .$$

$$\mathcal{C}=\left[\begin{array}{ccccc}0& 0& 0& 1& 1\\ 0& 0& 1& 0& 1\\ 0& 1& 1& 1& 0\end{array}\right].$$

$$R(t)={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\limsup_{n\to\infty}}\frac{\log_{2}A(n,t)}{n}.$$

$$A(n,t)\ge 2^{\lfloor \frac{n}{t}\rfloor }.$$

$$\sum_{h=1}^{t}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{j_{h}}\neq\sum_{l=1}^{t}\mathchoice{\mbox{\boldmath$\displaystyle c$}}{\mbox{\boldmath$\textstyle c$}}{\mbox{\boldmath$\scriptstyle c$}}{\mbox{\boldmath$\scriptscriptstyle c$}}_{k_{l}}.$$

$$e(G)\leq\left\{\begin{array}[]{ll}(2t-3)(v^{\frac{1}{2}}u^{\frac{t+2}{2t}}+u+v),&t\ \text{is\ even},\\ (2t-3)((uv)^{\frac{t+1}{2t}}+u+v),&t\ \text{is\ odd}.\end{array}\right.$$

$$A(n,t)\leq\left\{\begin{array}[]{ll}(2t-3)\big{(}2^{\frac{(t+2)n}{2t+2}+1}+2^{\frac{tn}{2t+2}}\big{)},&t\ \text{is\ even},\\ (2t-3)\big{(}2^{\frac{(t+1)n}{2t}}+2^{\lceil\frac{n}{2}\rceil}+2^{\lfloor\frac{n}{2}\rfloor}\big{)},&t\ \text{is\ odd}.\end{array}\right.$$

$$A(n,t)\leq\left\{\begin{array}[]{ll}(2t-3)(2^{\frac{t+2}{2t}n_{1}+\frac{n_{2}}{2}}+2^{n_{1}}+2^{n_{2}}),&t\ \text{is\ even},\\ (2t-3)(2^{\frac{t+1}{2t}n}+2^{n_{1}}+2^{n_{2}}),&t\ \text{is\ odd},\end{array}\right.$$

$$(n_{1},n_{2})=\left\{\begin{array}[]{ll}(\frac{tn}{2t+2},\frac{(t+2)n}{2t+2}),&t\ \text{is\ even},\\ (\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor),&t\ \text{is\ odd},\end{array}\right.$$

$$e(G)\le \frac{m-1}{\left(\genfrac{}{}{0ex}{}{p}{s-1}\right)}\left(\genfrac{}{}{0ex}{}{u}{s}\right)+v\frac{(p+1)(s-1)}{s}.$$

$$A(n,2)\leq\left\{\begin{array}[]{ll}2^{\lfloor\frac{2n}{3}\rfloor}+2^{\lceil\frac{n}{3}\rceil}(2^{\lceil\frac{n}{3}\rceil}-1)/2,&n\equiv 2\pmod{3},\\ 2^{\lceil\frac{2n}{3}\rceil}+2^{\lfloor\frac{n}{3}\rfloor}(2^{\lfloor\frac{n}{3}\rfloor}-1)/2,&\text{otherwise}.\end{array}\right.$$

$$A(n,2)\le 2^{n_2}+2^{n_1}(2^{n_1}-1)/2$$

$$(n_{1},n_{2})=\left\{\begin{array}[]{ll}(\lceil\frac{n}{3}\rceil,\lfloor\frac{2n}{3}\rfloor),&n\equiv 2\pmod{3},\\ (\lfloor\frac{n}{3}\rfloor,\lceil\frac{2n}{3}\rceil),&\text{otherwise}.\end{array}\right.$$

$$R(t)\leq\left\{\begin{array}[]{ll}\frac{1}{2}+\frac{1}{2t+2},&t\ \text{is\ even},\\ \frac{1}{2}+\frac{1}{2t},&t\ \text{is\ odd}.\end{array}\right.$$

$$R_{B}(t)={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\limsup_{n\to\infty}}\frac{\log_{2}B(n,t)}{n}.$$

$$\mathcal{C}=\left[\begin{array}{cccccc}1& 1& 0& 1& 1& 0\\ 1& 0& 1& 0& 0& 0\\ 0& 1& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 1\\ 0& 0& 0& 0& 1& 1\end{array}\right].$$

$$T(A)=\{z\in [n]:A\cup \{z\}\in \mathcal{F}\}.$$

$$3A(n,3,2)=\sum _{A\in \left(\genfrac{}{}{0ex}{}{[n]}{2}\right)}|T(A)|$$

Full text

I Introduction

The advancement of multimedia technologies with the development of communication networks has led to a tremendous use of multimedia content such as images, videos and so on. However, such an advantage also poses the challenging task of resisting unauthorized redistribution of multimedia content. Multimedia fingerprinting is a technique to protect continuous copyrighted data [39, 49] and several types of anti-collusion codes for multimedia fingerprinting have been investigated in recent decades, see [5, 10, 12, 22, 23, 31, 34] for example.

As in [39, 49], suppose that the multimedia content is represented as a real-valued vector \mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}=(x(1),x(2),\ldots, $x(m))\in\mathbb{R}^{m}$, called the host signal. To prevent unauthorized redistribution of $\textstyle x$ outside of $M$ authorized users, the dealer constructs a set of watermarks, also called fingerprints, using a linear modulation scheme based on $n\leq m$ noise-like orthonormal signals {\cal F}=\{\mathchoice{\mbox{\boldmath\displaystyle f}}{\mbox{\boldmath\textstyle f}}{\mbox{\boldmath\scriptstyle f}}{\mbox{\boldmath\scriptscriptstyle f}}_{i}\in\mathbb{R}^{m}:\ 1\leq i\leq n\}. The set ${\cal F}$ is known to the dealer but unknown to any user. The fingerprint \mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}_{j} of the $j$-th authorized user, $1\leq j\leq M$, is represented as

[TABLE]

where $c_{j}(i)\in\{-1,1\}$ for antipodal modulation and $c_{j}(i)\in\{0,1\}$ for on-off keying type of modulation [39]. In this paper, we concentrate on the on-off keying type of modulation, that is, $c_{j}(i)\in\{0,1\}$. Notice that there exists a one-to-one correspondence between \mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}_{j} and \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}=(c_{j}(1),c_{j}(2),\ldots,c_{j}(n))\in\{0,1\}^{n} due to the linear independence of \{\mathchoice{\mbox{\boldmath\displaystyle f}}{\mbox{\boldmath\textstyle f}}{\mbox{\boldmath\scriptstyle f}}{\mbox{\boldmath\scriptscriptstyle f}}_{i}:1\leq i\leq n\}. Hence, designing a collection of fingerprints with desired properties is equivalent to designing a set of binary vectors of length $n$, \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}:1\leq j\leq M\}\subseteq\{0,1\}^{n}, with the corresponding properties. Moreover, if for any $1\leq j\leq M$, \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j} has the same number of non-zero coordinates (i.e., constant-weight), then each user’s fingerprint defined by (1) is considered to have the same representation sparsity under {\cal F}=\{\mathchoice{\mbox{\boldmath\displaystyle f}}{\mbox{\boldmath\textstyle f}}{\mbox{\boldmath\scriptstyle f}}{\mbox{\boldmath\scriptscriptstyle f}}_{1},\mathchoice{\mbox{\boldmath\displaystyle f}}{\mbox{\boldmath\textstyle f}}{\mbox{\boldmath\scriptstyle f}}{\mbox{\boldmath\scriptscriptstyle f}}_{2},\ldots,\mathchoice{\mbox{\boldmath\displaystyle f}}{\mbox{\boldmath\textstyle f}}{\mbox{\boldmath\scriptstyle f}}{\mbox{\boldmath\scriptscriptstyle f}}_{n}\}.

In the above setting, the dealer distributes to the $j$-th authorized user the signal

[TABLE]

under the assumption that \|\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}\|\gg\|\mathchoice{\mbox{\boldmath\displaystyle w}}{\mbox{\boldmath\textstyle w}}{\mbox{\boldmath\scriptstyle w}}{\mbox{\boldmath\scriptscriptstyle w}}_{j}\|,111$\|\cdot\|$ is the Euclidean norm. which ensures that the embedded fingerprint does not make a significant change to the host signal. A group of malicious users (coalition) may come together and create a forged copy from their fingerprinted contents, but under the Multimedia Marking Assumption [23], they cannot manipulate the orthonormal signals \mathchoice{\mbox{\boldmath\displaystyle f}}{\mbox{\boldmath\textstyle f}}{\mbox{\boldmath\scriptstyle f}}{\mbox{\boldmath\scriptscriptstyle f}}_{i}, $1\leq i\leq n$. In a linear attack, a coalition $J\subseteq\{1,2,\ldots,M\}$ creates a forged copy \hat{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}} by taking a linear combination of their copies \mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}_{j} with some real-valued coefficients $\lambda_{j}$, that is,

[TABLE]

where $\sum_{j\in J}\lambda_{j}=1$, and $\lambda_{j}>0$ for all $j\in J$ since we consider that all users in the coalition (colluders) make contributions to the forged copy \hat{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}}. Equivalently,

[TABLE]

where the coefficients $\lambda_{j}$ are chosen by the coalition $J$ but unknown to the dealer. When $|J|=t$ and $\lambda_{j}=1/t$ for all $j\in J$, the linear attack is known as the averaging attack. Once a forged copy \hat{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}} is confiscated, the dealer can calculate

[TABLE]

for $1\leq k\leq n$ where $\langle\cdot,\cdot\rangle$ denotes the inner product, and obtain

[TABLE]

Obviously, \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}\in\mathbb{R}^{n} and $0\leq r(k)\leq 1$ for any $1\leq k\leq n$. To resist the linear attack, the dealer aims to identify the coalition $J$ based on the result $\textstyle r$ even without any knowledge of $\lambda_{j}$, $j\in J$, which requires that the set \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}:1\leq j\leq M\}\subseteq\{0,1\}^{n} has some desired properties.

In the previous work, Cheng and Miao [12] considered a discretized model of multimedia fingerprinting. In their setting, the discretization of the result $\textstyle r$ was considered, which can be represented by a mapping $\sigma$ on $\textstyle r$: \sigma(\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}})=\sigma(r(1))\times\cdots\times\sigma(r(n)) where

[TABLE]

for any $1\leq k\leq n$. Since $\sum_{j\in J}\lambda_{j}=1$ and $\lambda_{j}>0$ for all $j\in J$, by (3), it is easy to see that $\sigma(r(k))=\{c_{j}(k):j\in J\}$ for any $1\leq k\leq n$. Under this discretized model of multimedia fingerprinting, several types of codes have been proposed. In [12], Cheng and Miao introduced separable codes (SCs) to identify the coalition set $J$ based on \sigma(\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}), which were further investigated in [5, 22, 31] and more. It was proved in [12] that frameproof codes (FPCs), introduced by Boneh and Shaw in [7], have the ability of identifying the whole coalition set $J$ and the tracing algorithm based on an FPC is more efficient than that based on an SC. However the requirements of an FPC are stronger than those of an SC. To combine the advantages of FPCs and SCs, Jiang et al. [34] introduced strongly separable codes (SSCs) which lie between FPCs and SCs, and the tracing algorithm based on an SSC is as efficient as that based on an FPC. Also under the discretized model, Cheng et al. [10] considered the problem of identifying at least one colluder and introduced codes with the identifiable parent property for multimedia fingerprinting, whose largest code size and efficient decoding algorithm were later investigated in [32, 35] and more.

As can be seen, the above discretized model does not explore the randomness of $\lambda_{j}$ in the attack model (2). In this paper, we would like to go back to the original problem of designing \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}:1\leq j\leq M\}\subseteq\{0,1\}^{n} to trace back to the whole coalition $J$ based on the result $\textstyle r$ even though $\lambda_{j}$, $j\in J$ are unknown, that is, designing an anti-collusion code with complete traceability in multimedia fingerprinting, where the complete traceability refers to the ability of tracing back to all colluders, i.e., the whole coalition $J$.

In [23], Egorova et al. showed that this problem is in fact equivalent to the problem of designing a $t$-signature code for the weighted binary adder channel (WbAC). Suppose that $M$ users would like to communicate with the same destination through a shared WbAC in the multiple-access communication system, among which at most $t$ users are active simultaneously. To communicate successfully, each user $j$ is encoded to a unique vector \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in\{0,1\}^{n}, $1\leq j\leq M$. If the users in $J\subseteq[M]$ are active at the same time, they input their vectors simultaneously into the WbAC. The output in the destination is a vector $\textstyle r$ as in (4), where $\lambda_{j}$ plays the role of weight depending only on the channel but unknown to all encoders and the decoder. To identify all the active users in set $J$ using the corresponding channel output $\textstyle r$, it is required to design \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}:1\leq j\leq M\}\subseteq\{0,1\}^{n} with some desired properties. The case that $\lambda_{j}>0$, $j\in J$ are real numbers in WbAC was first investigated by Mathys [41]. Egorova et al. [23] first considered the scenario that $\lambda_{j}>0$, $j\in J$ are real numbers such that $\sum_{j\in J}\lambda_{j}=1$. Under this setting, they observed that the WbAC is essentially a modification of the multimedia fingerprinting channel, and the problem of designing \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}:1\leq j\leq M\}\subseteq\{0,1\}^{n} to identify the set of active users from the output $\textstyle r$ of WbAC is equivalent to the problem of designing \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}:1\leq j\leq M\}\subseteq\{0,1\}^{n} with complete traceability in multimedia fingerprinting where $\textstyle r$ is defined in (4). Based on this observation, they introduced the concept of signature codes for WbAC and multimedia fingerprinting and gave a direct construction for signature codes which yields a lower bound on the maximum size of $t$-signature codes. As far as we know, there are no other results on signature codes except that shown in [23].

In this paper, we would like to investigate signature codes from a combinatorial perspective. Specifically, we derive upper bounds on the maximum size of signature codes by combinatorial methods, which improve the known results in [23] for certain cases. We also investigate the combinatorial properties of signature codes with constant-weight and establish their optimal constructions. Moreover, we provide two constructions for signature codes which are equipped with efficient decoding algorithms. In addition, we consider some variants of signature codes for the noisy scenario as well. Due to the equivalence between WbAC and multimedia fingerprinting as described in the last paragraph, in the following contents, we will discuss codes mainly with respect to multimedia fingerprinting for simplicity.

The remainder of this paper is organized as follows. In Section II, we review the definition of signature codes and give an equivalent description of signature codes from a geometric viewpoint. Known results on signature codes and relationships between signature codes and $B_{t}$ codes are also presented. In Section III, we first relate $t$-signature codes to bipartite graphs without cycles of length less than or equal to $2t$ and then obtain general upper bounds for $t$-signature codes of length $n$ from the known results on bipartite graphs. Asymptotic upper bounds for $t$-signature codes are obtained based on the known results on $B_{t}$ codes. In Section IV, we first show that a $2$-signature code with constant-weight $2$ is equivalent to a $C_{4}$-free graph, yielding exact values of $A(n,2,2)$ and optimal constructions for $2$-signature codes with constant-weight $2$ for infinite values of $n$. Exact values of $A(n,3,2)$ are also derived and optimal constructions for $2$-signature codes with constant-weight $3$ are obtained for all $n$ with a few exceptions by investigating their combinatorial properties. In Section V, we provide two explicit constructions for $t$-signature codes and also investigate their decoding algorithms and applications. In Section VI, we show from a geometric viewpoint that there does not exist any binary code with complete traceability for noisy multimedia fingerprinting, and introduce a new type of signature codes with a weaker requirement which could be used in noisy multimedia fingerprinting. Conclusion is drawn in Section VII.

II Preliminaries

In this section, we first present the definition of signature code introduced in [23], and then give an equivalent description of signature codes from a geometric viewpoint. Known results on signature codes and relationships between signature codes and $B_{t}$ codes are also presented.

II-A Signature code

Let $n,M$ and $q$ be positive integers, and $Q=\{0,1,\ldots,q-1\}$ be the alphabet. A set {\cal C}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{M}\}\subseteq Q^{n} is called a $q$-ary code of length $n$ and size $M$, or $(n,M,q)$ code. Each \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}=(c_{j}(1),c_{j}(2),\ldots,c_{j}(n)) of ${\cal C}$ is called a codeword and $c_{j}(i)$ is called the $i$-th coordinate of \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}. Let ${\cal C}$ be an $(n,M,q)$ code. Define $R=(\log_{q}M)/n$ as the code rate of ${\cal C}$.

The matrix representation of ${\cal C}$ is an $n\times M$ matrix where each column vector is a codeword of ${\cal C}$. Denote $[n]=\{1,2,\ldots,n\}$. For any $I\subseteq[n]$ and \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}=(c(1),c(2),\ldots,c(n))\in{\cal C}, we denote \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I} as the punctured codeword of $\textstyle c$ restricted to $I$ by deleting all the coordinates $c(k)$ of $\textstyle c$ with $k\not\in I$, and denote {\cal C}|_{I}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I}:\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}\} as the punctured code of ${\cal C}$ restricted to $I$.

An $(n,M,2)$ code is also called a binary code. Let ${\cal C}$ be a binary code. For any codeword \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}, let \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}})=\{i\in[n]:c(i)=1\} be the support of $\textstyle c$. Denote $2^{[n]}$ as the family of all subsets of $[n]$, and $\binom{[n]}{w}$ as the family of all $w$-subsets of $[n]$. ${\cal C}$ has constant-weight $w$ if |\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}})|=w for any \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}.

The definition of signature code was stated in [23].

Definition II.1

([23])*

Let ${\cal C}$ be an $(n,M,2)$ code and $t\geq 2$ be an integer. ${\cal C}$ is an $(n,M,2)$ $t$-signature code if*

for any two distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, we have

[TABLE]

for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1.

We first generalize the definition of signature code by considering multi-subsets of ${\cal C}$ in the condition of Definition II.1. A multi-set is a collection of elements which allows for multiple occurrences of the elements. The number of occurrence of each element in a multi-set is called the multiplicity of this element. The size of a multi-set is the sum of the multiplicities of all its elements. For any multi-set $S$, we call the set formed by all the elements in $S$ as the base set of $S$. Two multi-sets are distinct if their base sets are distinct, or there exists an element such that the multiplicity of this element in two multi-sets are different. A multi-subset $S_{1}$ of a set $S_{2}$ is a multi-set such that the base set of $S_{1}$ is a subset of $S_{2}$.

Example 1

Let $S=\{1,2,3,4,5\}$ be a set. Then $S_{1}=\{1,1,2\}$ and $S_{2}=\{1,2,2\}$ are two distinct multi-subsets of $S$ with the same base set $\{1,2\}$. Let $S_{3}=\{1,2,2,3\}$. Then the base set of $S_{3}$ is $\{1,2,3\}$ which is distinct to that of $S_{1}$ and $S_{2}$.

We have the following equivalent but more general description of Definition II.1.

Proposition II.1

An $(n,M,2)$ code ${\cal C}$ is a $t$-signature code if and only if for any two multi-subsets \widehat{{\cal C}}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{s}}\} and \widehat{{\cal C}}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{m}}\} of ${\cal C}$ with $s,m\leq t$ and distinct base sets, we have

[TABLE]

for any real numbers $\lambda_{j_{h}},\lambda_{k_{l}}^{\prime}>0$ such that $\sum_{h=1}^{s}\lambda_{j_{h}}=\sum_{l=1}^{m}\lambda_{k_{l}}^{\prime}=1$.

Proof:

The sufficiency could be easily derived by Definition II.1. To show the necessity, we assume that {\cal C}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\cdots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{M}\} is a $t$-signature code, but there exist two multi-subsets \widehat{{\cal C}}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{s}}\},\ \widehat{{\cal C}}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{m}}\} of ${\cal C}$ with $s,m\leq t$ and distinct base sets, and also exist $\lambda_{j_{h}},\lambda_{k_{l}}^{\prime}>0$ with $\sum_{h=1}^{s}\lambda_{j_{h}}=\sum_{l=1}^{m}\lambda_{k_{l}}^{\prime}=1$ such that

[TABLE]

By combining like terms in the left-hand and right-hand sides of (5) respectively, we can obtain

[TABLE]

where ${\cal C}_{1},{\cal C}_{2}$ are base sets of $\widehat{{\cal C}}_{1},\widehat{{\cal C}}_{2}$ respectively satisfying $|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$ and ${\cal C}_{1}\neq{\cal C}_{2}$, and $\lambda_{j},\lambda^{\prime}_{k}>0$ are real numbers satisfying \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1. This is a contradiction to the assumption that ${\cal C}$ is a $t$-signature code (see Definition II.1). The conclusion follows.

∎

Proposition II.1 shows that in Definition II.1, the condition “two distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$” could be equivalently replaced by “two multi-subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ of size no more than $t$ with distinct bases sets”. We remark that in the remainder of this paper, ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ always refer to subsets unless otherwise specified.

II-B A geometric perspective

Signature codes can be interpreted geometrically. Let $\{0,1\}^{n}$ be the vertex-set of an $n$-dimensional hypercube. Then a binary code of length $n$ corresponds to a subset of vertices of an $n$-dimensional hypercube. For any subset of vertices {\cal C}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{m}\}\subseteq\{0,1\}^{n}, define the open convex polytope of ${\cal C}$ as

[TABLE]

For any distinct ${\cal C}_{1},{\cal C}_{2}\subseteq\{0,1\}^{n}$, the open convex polytopes of ${\cal C}_{1}$ and ${\cal C}_{2}$ do not cross if

[TABLE]

Then we have

Proposition II.2

An $(n,M,2)$ code ${\cal C}$ is a $t$-signature code if and only if for any distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, we have $\mathcal{P}({\cal C}_{1})\cap\mathcal{P}({\cal C}_{2})=\emptyset$.

Proposition II.2 can be immediately derived from Definition II.1. The following example illustrates our argument.

Example 2

Consider the vertices of the $3$-dimensional hypercube as shown in Figure 1. The open convex polytope formed by any two distinct vertices $v_{i},v_{j}$ is the line segment between $v_{i}$ and $v_{j}$ except $v_{i},v_{j}$ themselves. Let ${\cal H}_{0}=\{v_{1},v_{2},v_{3},v_{6},v_{8}\}$. It is easy to check that the open convex polytopes formed by any two vertices in ${\cal H}_{0}$ do not cross.

Then the set of vectors ${\cal C}$ corresponding to ${\cal H}_{0}$, that is, ${\cal C}=\{(0,0,0),(0,0,1),(0,1,1),(1,0,1),(1,1,0)\}$, is a $(3,5,2)$ $2$-signature code. The matrix representation of ${\cal C}$ is presented below.

[TABLE]

In the application of an $(n,M,2)$ $t$-signature code ${\cal C}$ to multimedia fingerprinting, suppose that ${\cal C}_{1}$ corresponds to the set of fingerprints distributed to a coalition set $J$. As introduced in Section I, once a forged copy produced by the coalition $J$ is confiscated, the dealer can calculate a result $\textstyle r$. By (3), (4) and (6), $\textstyle r$ is a point in ${\cal P}({\cal C}_{1})$. Then we can trace back to ${\cal C}_{1}$, or equivalently the coalition set $J$, by checking that to which open convex polytope, formed by a subset of ${\cal C}$ with size no more than $t$, $\textstyle r$ belongs. The computational complexity of the tracing algorithm is $O(nM^{t})$.

II-C Known results

Let $A(n,t)$ denote the maximum size of a $t$-signature code of length $n$, and $A(n,w,t)$ denote the maximum size of a $t$-signature code of length $n$ and constant-weight $w$. A $t$-signature code of length $n$ is called optimal if it has size $A(n,t)$, and a $t$-signature code of length $n$ and constant-weight $w$ is called optimal if it has size $A(n,w,t)$. Denote the largest asymptotic code rate of $t$-signature codes as

[TABLE]

In [23], Egorova et al. found that a binary code with any $2t$ codewords linearly independent is a $t$-signature code. Based on this observation, they constructed signature codes from binary Goppa codes in coding theory and obtained a lower bound on $A(n,t)$.

Theorem II.1

([23])**

[TABLE]

This immediately implies

Corollary II.2

Let $t\geq 2$ be an integer. Then $R(t)\geq 1/t.$

In the literature, the exact values of $A(n,t)$ and $R(t)$ are generally unknown except the lower bounds in Theorem II.1 and Corollary II.2. However, the lower bound on $A(n,t)$ in Theorem II.1 is not tight in general since for instance it shows $A(3,2)\geq 2$, while Example 2 implies $A(3,2)\geq 5$. The main purpose of this paper is to explore bounds on $A(n,t)$ and $A(n,w,t)$ by using combinatorial methods, and to provide constructions for $t$-signature codes which are equipped with efficient tracing algorithms.

II-D Relationships with $B_{t}$ codes

In this subsection, we build relationships between signature codes and $B_{t}$ codes.

The investigation of $B_{t}$ code was motivated by $B_{t}$-sequence introduced by Erdős and Turán [26]. $B_{t}$ codes for $t=2$ were first studied by Lindström in [37] and later in [38] under the name of $B_{2}$-sequences of vectors. In recent decades, $B_{t}$ codes were investigated due to their applications to the multiple access adder channel. The interested reader is referred to [17, 19, 20, 21, 33] for more details.

Definition II.2

An $(n,M,q)$ code ${\cal C}$ is a $B_{t}$ code, or $(n,M,q)$ $B_{t}$ code, if for any two distinct multi-subsets \widehat{{\cal C}}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{t}}\} and \widehat{{\cal C}}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{t}}\} of ${\cal C}$, we have

[TABLE]

From Definition II.2, it is obvious that when $q=2$ and $t=2$, an $(n,M,2)$ code ${\cal C}$ is $B_{2}$ code if and only if \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\neq\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s} for any distinct \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\in{\cal C}. We have the following observation on the relationship between signature codes and $B_{t}$ codes.

Lemma II.1

Let ${\cal C}$ be an $(n,M,2)$ code and $t\geq 2$ be an integer. If ${\cal C}$ is a $t$-signature code, then ${\cal C}$ is a $B_{t}$ code.

Proof:

Let {\cal C}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{M}\} be an $(n,M,2)$ code. Assume that ${\cal C}$ is a $t$-signature code, but not a $B_{t}$ code. By Definition II.2, there exist two distinct multi-subsets \widehat{{\cal C}}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{t}}\} and \widehat{{\cal C}}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{1}},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{t}}\} of ${\cal C}$ such that \sum_{h=1}^{t}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{h}}=\sum_{l=1}^{t}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{l}}. By subtracting all the common codewords counted in the two summations, we can obtain a non-empty multi-subset \widehat{{\cal A}}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{1}}^{\prime},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{m}}^{\prime}\} of $\widehat{{\cal C}}_{1}$ and a non-empty multi-subset \widehat{{\cal A}}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{1}}^{\prime},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{m}}^{\prime}\} of $\widehat{{\cal C}}_{2}$ which satisfy $m\leq t$, \sum_{h=1}^{m}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{h}}^{\prime}=\sum_{l=1}^{m}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{l}}^{\prime} and the base sets of $\widehat{{\cal A}}_{1},\widehat{{\cal A}}_{2}$ are distinct. Let $\lambda_{j_{h}}=\lambda_{k_{l}}=1/m$ for any $1\leq h,l\leq m$. Then we have $\sum_{h=1}^{m}\lambda_{j_{h}}=\sum_{l=1}^{m}\lambda_{k_{l}}=1$. Furthermore, we have \sum_{h=1}^{m}\lambda_{j_{h}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j_{h}}^{\prime}=\sum_{l=1}^{m}\lambda_{k_{l}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k_{l}}^{\prime}, a contradiction to the assumption that ${\cal C}$ is a $t$-signature code by Proposition II.1. The lemma follows. ∎

The converse of Lemma II.1 does not hold in general. However, when $t=2$, we have the following observation.

Lemma II.2

Let ${\cal C}$ be an $(n,M,2)$ code. Then ${\cal C}$ is a $2$-signature code if and only if \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\neq\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s} for any distinct \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\in{\cal C}. In other words, ${\cal C}$ is a $2$-signature code if and only if it is a $B_{2}$ code.

Proof:

The necessity is clear from Lemma II.1. Now we show the sufficiency. Assume that for any distinct \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\in{\cal C}, \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\neq\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}, but ${\cal C}$ is not a $2$-signature code. Then according to Definition II.1, there exist two distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq 2$, and also exist $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k} where \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1. We discuss the following cases.

If $|{\cal C}_{1}|=1$ or $|{\cal C}_{2}|=1$, then we must have ${\cal C}_{1}={\cal C}_{2}$, a contradiction to the assumption. 2. 2.

If $|{\cal C}_{1}|=|{\cal C}_{2}|=2$ and ${\cal C}_{1}\cap{\cal C}_{2}\neq\emptyset$, then we can have ${\cal C}_{1}={\cal C}_{2}$, a contradiction. 3. 3.

If $|{\cal C}_{1}|=|{\cal C}_{2}|=2$ and ${\cal C}_{1}\cap{\cal C}_{2}=\emptyset$, without loss of generality, we may assume that {\cal C}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2}\} and {\cal C}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}\}. According to the assumption, there exist $\lambda_{1},\lambda_{2},\lambda_{3}^{\prime},\lambda_{4}^{\prime}>0$ such that \lambda_{1}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1}+\lambda_{2}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2}=\lambda_{3}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3}+\lambda_{4}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4} where $\lambda_{1}+\lambda_{2}=\lambda_{3}^{\prime}+\lambda_{4}^{\prime}=1$. For any $1\leq i\leq n$, we discuss the following three subcases of $\{c_{1}(i),c_{2}(i)\}$.

1. 3.a)

If $\{c_{1}(i),c_{2}(i)\}=\{0\}$, then $\lambda_{1}c_{1}(i)+\lambda_{2}c_{2}(i)=0=\lambda_{3}^{\prime}c_{3}(i)+\lambda_{4}^{\prime}c_{4}(i)$, which implies $c_{3}(i)=c_{4}(i)=0$ since $\lambda_{3}^{\prime},\lambda_{4}^{\prime}>0$. 2. 3.b)

If $\{c_{1}(i),c_{2}(i)\}=\{1\}$, then $\lambda_{1}c_{1}(i)+\lambda_{2}c_{2}(i)=1=\lambda_{3}^{\prime}c_{3}(i)+\lambda_{4}^{\prime}c_{4}(i)$ since $\lambda_{1}+\lambda_{2}=1$, which implies $c_{3}(i)=c_{4}(i)=1$ since $\lambda_{3}^{\prime},\lambda_{4}^{\prime}>0$ and $\lambda_{3}^{\prime}+\lambda_{4}^{\prime}=1$. 3. 3.c)

If $\{c_{1}(i),c_{2}(i)\}=\{0,1\}$, then $0<\lambda_{1}c_{1}(i)+\lambda_{2}c_{2}(i)=\lambda_{3}^{\prime}c_{3}(i)+\lambda_{4}^{\prime}c_{4}(i)<1$, which implies $\{c_{3}(i),c_{4}(i)\}=\{0,1\}$.

Then for case 3), we have \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}, also a contradiction to the assumption since \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2}\}\cap\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}\}=\emptyset.

The conclusion follows. ∎

III Upper bounds on $t$-signature codes

In this section, we investigate upper bounds on $A(n,t)$ and $R(t)$. First, we derive a general upper bound on $A(n,t)$ by connecting a $t$-signature code with a bipartite graph containing no cycles of length less than or equal to $2t$, which also implies an upper bound on $R(t)$. Second, we give an improved upper bound for $R(t)$ based on the known results on $B_{t}$ codes.

III-A General upper bounds on $t$-signature codes

Denote $G=(V_{1},V_{2},E)$ as a bipartite graph where $V_{1},V_{2}$ are two disjoint sets of vertices and $E$ is the set of edges each connecting one vertex in $V_{1}$ and the other in $V_{2}$. $G=(V_{1},V_{2},E)$ is called complete if every vertex in $V_{1}$ is connected to every vertex in $V_{2}$. A complete bipartite graph $G=(V_{1},V_{2},E)$ with $|V_{1}|=m_{1}$ and $|V_{2}|=m_{2}$ is denoted as $K_{m_{1},m_{2}}$. A cycle $C_{k}$ of $G=(V_{1},V_{2},E)$ is a sequence of vertices $v_{1}v_{2}\cdots v_{k}v_{1}$, where $v_{i}$, $1\leq i\leq k$ are all distinct and $\{v_{i},v_{i+1}\},\{v_{k},v_{1}\}\in E$ for any $1\leq i\leq k-1$. The number of vertices in a cycle is called the length of this cycle, and the girth of a graph $G$ is the length of the shortest cycle in $G$. If a graph $G$ contains no cycle of length $k$, it is called $C_{k}$-free. A matching of $G$ is a subset of edges without common vertices, and a perfect matching is a matching which matches all vertices in $G$.

Let ${\cal C}$ be an $(n,M,2)$ code. For a partition of $[n]=I_{1}\cup I_{2}$ with $|I_{1}|=n_{1}$, $|I_{2}|=n_{2}$ and $n_{1}+n_{2}=n$, we define a bipartite graph corresponding to ${\cal C}$ as $G=(V_{1},V_{2},E)$ such that $V_{1}=\{0,1\}^{n_{1}}$, $V_{2}=\{0,1\}^{n_{2}}$ and for any \mathchoice{\mbox{\boldmath\displaystyle u}}{\mbox{\boldmath\textstyle u}}{\mbox{\boldmath\scriptstyle u}}{\mbox{\boldmath\scriptscriptstyle u}}\in V_{1} and \mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}\in V_{2}, \{\mathchoice{\mbox{\boldmath\displaystyle u}}{\mbox{\boldmath\textstyle u}}{\mbox{\boldmath\scriptstyle u}}{\mbox{\boldmath\scriptscriptstyle u}},\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}\}\in E if and only if there exists a codeword \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C} such that \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I_{1}}=\mathchoice{\mbox{\boldmath\displaystyle u}}{\mbox{\boldmath\textstyle u}}{\mbox{\boldmath\scriptstyle u}}{\mbox{\boldmath\scriptscriptstyle u}} and \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I_{2}}=\mathchoice{\mbox{\boldmath\displaystyle v}}{\mbox{\boldmath\textstyle v}}{\mbox{\boldmath\scriptstyle v}}{\mbox{\boldmath\scriptscriptstyle v}}. Denote $e(G)$ as the number of edges in $G$. It is obvious that $|V_{1}|=2^{n_{1}}$, $|V_{2}|=2^{n_{2}}$ and $e(G)=|{\cal C}|$. Moreover, we have the following observation.

Lemma III.1

If ${\cal C}$ is a $t$-signature code, then for any partition of $[n]=I_{1}\cup I_{2}$, the girth of the corresponding bipartite graph $G=(V_{1},V_{2},E)$ is at least $2t+2$.

Proof:

Note that a bipartite graph contains no cycles of odd length. To show $G=(V_{1},V_{2},E)$ has girth at least $2t+2$, it suffices to show that $G=(V_{1},V_{2},E)$ contains no cycles of length $2k$ for any $2\leq k\leq t$. Assume that ${\cal C}$ is a $t$-signature code, but $G=(V_{1},V_{2},E)$ contains a cycle of length $2k$ for some $2\leq k\leq t$. Without loss of generality, we may assume that the cycle $C_{2k}$ is $v_{1}v_{2}\ldots v_{2k-1}v_{2k}v_{1}$, where $v_{1},v_{3},\ldots,v_{2k-1}\in V_{1}$ and $v_{2},v_{4},\ldots,v_{2k}\in V_{2}$. Let $e_{i}=\{v_{i},v_{i+1}\}$ for $1\leq i\leq 2k-1$, and $e_{2k}=\{v_{2k},v_{1}\}$. Then $\{e_{i}:1\leq i\leq 2k,\ i\ \text{is\ odd}\}$ and $\{e_{i}:1\leq i\leq 2k,\ i\ \text{is\ even}\}$ are two disjoint perfect matchings of the cycle $C_{2k}$. Recall that each edge of $G$ corresponds to a codeword of ${\cal C}$. Let \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}\in{\cal C} be the corresponding codeword of $e_{i}$ in $C_{2k}$ for $1\leq i\leq 2k$, and {\cal C}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}:1\leq i\leq 2k,\ i\ \text{is\ odd}\}, {\cal C}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}:1\leq i\leq 2k,\ i\ \text{is\ even}\}. Then ${\cal C}_{1}$ and ${\cal C}_{2}$ are two disjoint subsets of ${\cal C}$ such that $|{\cal C}_{1}|=|{\cal C}_{2}|=k$, \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}_{1}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I_{1}}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}_{2}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I_{1}}=\sum_{i\ \text{is odd}}v_{i} and \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}_{1}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I_{2}}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}_{2}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I_{2}}=\sum_{i\ \text{is even}}v_{i}. Therefore, we have \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}_{1}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}_{2}}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}, a contradiction to Lemma II.1 since ${\cal C}$ is a $t$-signature code. The conclusion follows. ∎

The following result on bipartite graphs was shown in [42].

Lemma III.2

([42])* Let $G=(V_{1},V_{2},E)$ be a bipartite graph with $|V_{1}|=u$ and $|V_{2}|=v$. If $G$ is $C_{2t}$-free, then*

[TABLE]

By Lemmas III.1 and III.2, we obtain

Theorem III.1

Let $n,t\geq 2$ be integers. Then

[TABLE]

Proof:

Let ${\cal C}$ be a $t$-signature code of length $n$ and size $M=A(n,t)$. By Lemma III.1, for any partition of $[n]=I_{1}\cup I_{2}$ with $|I_{1}|=n_{1}$, $|I_{2}|=n_{2}$ and $n_{1}+n_{2}=n$, the corresponding bipartite graph $G=(V_{1},V_{2},E)$ with $|V_{1}|=2^{n_{1}}$, $|V_{2}|=2^{n_{2}}$ and $e(G)=A(n,t)$ is $C_{2k}$-free for any $2\leq k\leq t$. Then by Lemma III.2, we have that

[TABLE]

holds for any integers $n_{1},n_{2}$ such that $n_{1},n_{2}\geq 0$ and $n_{1}+n_{2}=n$. To make the upper bound on $A(n,t)$ as small as possible, we take

[TABLE]

and then (7) follows.

∎

We mention that the above approach of upper bounding the size of a code by exploring its connection with graphs without small cycles was used in [10, 11, 18] as well. When $t=2$, Theorem III.1 shows that $A(n,2)\leq 2\cdot 2^{\frac{2n}{3}}+2^{\frac{n}{3}}$. A better upper bound for $A(n,2)$ could be obtained by using Roman’s bound [44] on bipartite graphs.

Lemma III.3

([44])* Let $G=(V_{1},V_{2},E)$ be a bipartite graph with $|V_{1}|=u$, $|V_{2}|=v$, and let $p$ be an integer such that $p\geq s-1$. If G contains no $K_{s,m}$, then*

[TABLE]

By Lemmas III.1 and III.3, we have

Theorem III.2

Let $n\geq 2$ be an integer. Then

[TABLE]

Proof:

Let ${\cal C}$ be a $2$-signature code of length $n$ and size $M=A(n,2)$. Similarly, according to Lemma III.1, for any partition of $[n]=I_{1}\cup I_{2}$ with $|I_{1}|=n_{1}$, $|I_{2}|=n_{2}$ and $n_{1}+n_{2}=n$, the corresponding bipartite graph $G=(V_{1},V_{2},E)$ with $|V_{1}|=2^{n_{1}}$, $|V_{2}|=2^{n_{2}}$ and $e(G)=A(n,2)$ is $C_{4}$-free. Note that $C_{4}\cong K_{2,2}$. Then by choosing $s=m=2$ and $p=1$ in Lemma III.3, we have that

[TABLE]

holds for any integers $n_{1},n_{2}$ such that $n_{1},n_{2}\geq 0$ and $n_{1}+n_{2}=n$. To make the upper bound on $A(n,2)$ as small as possible, we take

[TABLE]

Then (8) follows.

∎

It can be checked that the upper bound on $A(n,2)$ in Theorem III.2 is tighter than that in Theorem III.1 for the case $t=2$. Indeed, for instance Theorem III.1 tells $A(3,2)\leq 10$ while Theorem III.2 shows $A(3,2)\leq 5$. The following result shows that the upper bound on $A(n,2)$ in Theorem III.2 is tight for certain cases.

Corollary III.3

$A(2,2)=3$*, $A(3,2)=5$ and $A(4,2)=7$. *

Proof:

For $n=2$, it is easy to check that ${\cal C}=\{(1,0),(0,1),(1,1)\}$ is a $2$-signature code. Moreover, Theorem III.2 shows that $A(2,2)\leq 3$, resulting $A(2,2)=3$.

For $n=3$, Theorem III.2 shows that $A(3,2)\leq 5$ which, together with Example 2, yields $A(3,2)=5$.

For $n=4$, we have that ${\cal C}=\{(1,0,0,0),(0,1,0,0),(0,0,1,0),(1,1,0,0),(1,1,1,0),(0,0,0,1),(1,1,1,1)\}$ is a $2$-signature code, implying $A(4,2)\geq 7$. Notice that Theorem III.2 only tells that $A(4,2)\leq 9$. However, we can show $A(4,2)\leq 7$ whose proof is deferred to Appendix, and thus we have $A(4,2)=7$. ∎

From Theorem III.1, we can obtain an upper bound on $R(t)$ for any integer $t\geq 2$. Here, we remark that although the upper bound on $A(n,2)$ in Theorem III.2 is tighter than that in Theorem III.1 for $t=2$, they imply the same upper bound on the largest asymptotic code rate $R(2)\leq 2/3$. In general, by Theorem III.1, we have

Corollary III.4

Let $t\geq 2$ be an integer. Then

[TABLE]

III-B Asymptotic upper bounds on $t$-signature codes

In this subsection, we provide an improved upper bound on $R(t)$ based on Lemma II.1 and the known results on $B_{t}$ codes.

Denote $B(n,t)$ as the maximum size of a binary $B_{t}$ code of length $n$. Define the largest asymptotic code rate of binary $B_{t}$ codes as

[TABLE]

Denote H_{t}=-\sum_{k=0}^{t}\binom{t}{k}2^{-t}\log_{2}\big{(}\binom{t}{k}2^{-t}\big{)} and $h_{t}=\log_{2}(t+1)+t/(t+1)$. The best known upper bounds on $R_{B}(t)$ are as follows.

Lemma III.4

Let $t\geq 2$ be an integer. Then

([14, 21])* $R_{B}(2)\leq 0.5753$ and $R_{B}(4)\leq 0.4451$.* 2. 2.

([19, 20])* R_{B}(2t-1)\leq\left\{\begin{array}[]{l}(t/H_{t}+(t-1)/h_{t})^{-1},\ 2\leq t\leq 5,\\ H_{2t-1}/(2t-1),\ t\geq 6.\end{array}\right.* 3. 3.

([19, 20])* R_{B}(2t)\leq\left\{\begin{array}[]{l}(t/H_{t}+t/h_{t})^{-1},\ 3\leq t\leq 5,\\ H_{2t}/(2t),\ t\geq 6.\end{array}\right.* 4. 4.

([17])* For sufficiently large $t$, $R_{B}(t)\leq(\log_{2}t+\log_{2}(\pi e/2))/(2t)+O(1/t^{3})$.*

It is obvious from Lemmas II.1 and II.2 that $R(t)\leq R_{B}(t)$ for any integer $t\geq 2$ and the equality holds for $t=2$.

Combining this with Lemma III.4, we obtain

Theorem III.5

Let $t\geq 2$ be an integer. Then

$R(2)\leq 0.5753$* and $R(4)\leq 0.4451$.* 2. 2.

R(2t-1)\leq\left\{\begin{array}[]{l}(t/H_{t}+(t-1)/h_{t})^{-1},\ 2\leq t\leq 5,\\ H_{2t-1}/(2t-1),\ t\geq 6.\end{array}\right.** 3. 3.

R(2t)\leq\left\{\begin{array}[]{l}(t/H_{t}+t/h_{t})^{-1},\ 3\leq t\leq 5,\\ H_{2t}/(2t),\ t\geq 6.\end{array}\right.** 4. 4.

For sufficiently large integer $t$, $R(t)\leq(\log_{2}t+\log_{2}(\pi e/2))/(2t)+O(1/t^{3})$.

We remark that the upper bound of $R(t)$ obtained in Theorem III.5 is tighter than that obtained in Corollary III.4. For example, Corollary III.4 implies $R(2)\leq 2/3$ and $R(4)\leq 0.6$, while Theorem III.5 shows that $R(2)\leq 0.5753$ and $R(4)\leq 0.4451$.

IV Optimal $2$-signature codes with constant-weights $2$ and $3$

In this section, we study the combinatorial properties of $2$-signature codes of length $n$ and constant-weights $2$ and $3$, respectively. Accordingly, bounds on $A(n,2,2)$ and $A(n,3,2)$ are obtained, and optimal $2$-signature codes of length $n$ and constant-weights $2$ and $3$ for infinite values of $n$ are constructed respectively.

IV-A Optimal $2$-signature codes with constant-weight $2$

Let ${\cal C}$ be a binary code of length $n$ and constant-weight $2$. Let $G=(V,E)$ be the corresponding graph of ${\cal C}$ where $V=[n]$ and for any distinct $i,j\in[n]$, $\{i,j\}\in E$ if and only if there exists a codeword \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C} such that \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}})=\{i,j\}. Then we have the following observation.

Lemma IV.1

${\cal C}$* is a $2$-signature code of constant-weight $2$ if and only if its corresponding graph $G$ is $C_{4}$-free.*

Proof:

We first show the necessity. If there exist four vertices $i,j,k,s\in[n]$ which form a cycle $ijksi$ in graph $G$, then there must exist four distinct codewords \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}\in{\cal C} such that \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1})=\{i,j\}, \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2})=\{j,k\}, \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3})=\{k,s\} and \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4})=\{s,i\}. It is easy to verify that \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}, a contradiction to Lemma II.1.

On the other hand, assume that the corresponding graph $G$ of ${\cal C}$ is $C_{4}$-free, but ${\cal C}$ is not a $2$-signature code. By Lemma II.1, there must exist four distinct codewords \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\in{\cal C} such that \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}. If \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i})\cap\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j})\neq\emptyset, then we have \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\}, a contradiction to the assumption. Thus there must be four elements in \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i})\cup\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j})=\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k})\cup\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}) which form a cycle of length $4$ in $G$, a contradiction to the assumption. The lemma follows. ∎

Lemma IV.1 shows that if there exists a $2$-signature code with constant-weight $2$, then we can construct a $C_{4}$-free graph, and conversely, given a $C_{4}$-free graph, we can obtain a $2$-signature code with constant-weight $2$. The following example illustrates this relationship.

Example 3

Consider the $C_{4}$-free graph on $5$ vertices depicted in Figure 2.

Then according to Lemma IV.1, we can obtain a $2$-signature code ${\cal C}$ of length $5$ and constant-weight $2$ as follows.

[TABLE]

Let $g(n;C_{4})$ denote the maximum number of edges in a $C_{4}$-free graph with $n$ vertices. Then by Lemma IV.1, we have $A(n,2,2)=g(n;C_{4})$. It was proved in [6, 25] that $g(n;C_{4})$ is asymptotically equal to $(1/2+o(1))n^{3/2}$. In [28, 29, 30], it was shown that if $n=m^{2}+m+1$ and $m$ is a power of $2$ or a prime power greater than $13$, then $g(n;C_{4})=m(m+1)^{2}/2$ and a $C_{4}$-free graph with $n$ vertices and $m(m+1)^{2}/2$ edges could be constructed from the polarity of a projective plane of order $m$, see [25, 28, 29, 30] for more details. For any integer $2\leq n\leq 21$, the exact value of $g(n;C_{4})$ and the corresponding $C_{4}$-free graph with $n$ vertices and $g(n;C_{4})$ edges were presented in [13]. Based on these known results for $C_{4}$-free graphs and Lemma IV.1, we can obtain the following results for $2$-signature codes with constant-weight $2$.

Theorem IV.1

Let $n\geq 2$ be an integer.

$A(n,2,2)$* is asymptotically equal to $(1/2+o(1))n^{3/2}$.* 2. 2.

If $n=m^{2}+m+1$ where $m$ is a power of $2$ or $m>13$ is a prime power, then $A(n,2,2)=m(m+1)^{2}/2$ and an optimal $2$-signature code of length $n$ and constant-weight $2$ can be constructed from the polarity of a projective plane of order $m$. 3. 3.

For $2\leq n\leq 21$, the exact values of $A(n,2,2)$ are listed in Table I, and optimal $2$-signature codes of length $n$ and constant-weight $2$ can be constructed from the corresponding $C_{4}$-free graphs presented in **[13]**.

We remark that Theorem IV.1 is in fact a consequence of the combination of Lemma IV.1 and the known results for $C_{4}$-free graphs outlined above. Besides, the $C_{4}$-free graph depicted in Figure 2 was presented in [13]. By Theorem IV.1, the obtained $(5,6,2)$ $2$-signature code ${\cal C}$ with constant-weight $2$ in Example 3 is optimal.

IV-B Optimal $2$-signature codes with constant-weight $3$

In this subsection, we investigate the combinatorial properties of $2$-signature codes of constant-weight $3$ by means of Frankl and Füredi’s method in [27], where they studied binary codes using techniques in extremal set theory.

Let ${\cal C}$ be a binary code of length $n$ and constant-weight $3$. Notice that ${\cal C}$ can be uniquely described as a family ${\cal F}\subseteq\binom{[n]}{3}$ where {\cal F}=\{\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}):\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}\}. For any $A\in\binom{[n]}{2}$, denote

[TABLE]

Then we have the following observation.

Lemma IV.2

${\cal C}$* is a $2$-signature code of constant-weight $3$ if and only if $|T(A)\cap T(A^{\prime})|\leq 1$ for any distinct $A,A^{\prime}\in\binom{[n]}{2}$.*

Proof:

To show the necessity, assume that there exist distinct $A,A^{\prime}\in\binom{[n]}{2}$ and also distinct elements $z,z^{\prime}\in[n]$ such that $\{z,z^{\prime}\}\subseteq T(A)\cap T(A^{\prime})$. Then $A\cup\{z\},A\cup\{z^{\prime}\},A^{\prime}\cup\{z\},A^{\prime}\cup\{z^{\prime}\}$ are four distinct members of ${\cal F}$. Suppose that their corresponding codewords in ${\cal C}$ are \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}, respectively. Then we have \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{3}, a contradiction to Lemma II.1.

Next, we show the sufficiency. Assume that ${\cal C}$ satisfies that $|T(A)\cap T(A^{\prime})|\leq 1$ for any distinct $A,A^{\prime}\in\binom{[n]}{2}$, but is not a $2$-signature code. Then according to Lemma II.1, there exist distinct codewords \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\in{\cal C} such that \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}. Let B_{m}=\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{m}), $m\in\{i,j,k,s\}$ which are four distinct members of ${\cal F}$. Suppose that $B_{i}=\{a,b,c\}$ and $B_{j}=\{x,y,z\}$. If $|B_{i}\cap B_{j}|=2$, then we must have $\{B_{i},B_{j}\}=\{B_{k},B_{s}\}$ which implies \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\}, a contradiction to the assumption. So, we have $|B_{i}\cap B_{j}|\leq 1$. Now we discuss the following two cases.

If $|B_{i}\cap B_{j}|=1$, assume that $c=x$. Then $c\in B_{k}\cap B_{s}$. Recall that \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s} and $\{B_{i},B_{j}\}\cap\{B_{k},B_{s}\}=\emptyset$. Thus one of $a$ and $b$ should be in $B_{k}$ and the other in $B_{s}$, and so as $y$ and $z$. Without loss of generality, we may assume that $B_{k}=\{a,y,c\}$ and $B_{s}=\{b,z,c\}$. Let $A=\{a,c\}$ and $A^{\prime}=\{z,c\}$. Then $\{b,y\}\in\binom{T(A)}{2}\cap\binom{T(A^{\prime})}{2}$, a contradiction to that $|T(A)\cap T(A^{\prime})|\leq 1$. 2. 2.

If $|B_{i}\cap B_{j}|=0$, then $B_{k}\cap B_{s}=\emptyset$. Without loss of generality, we may assume that $B_{k}=\{a,b,x\}$ and $B_{s}=\{c,y,z\}$. Let $A=\{a,b\}$ and $A^{\prime}=\{y,z\}$. Then $\{c,x\}\in\binom{T(A)}{2}\cap\binom{T(A^{\prime})}{2}$, also a contradiction.

The conclusion then follows. ∎

Based on Lemma IV.2, we shall derive an upper bound for $A(n,3,2)$ which is the same as an upper bound for weakly union-free families given by Frankl and Füredi [27]. However, the bound cannot be obtained directly from the relationship between signature codes and weakly union-free families, which will be shown in Lemma IV.3 afterwards.

Theorem IV.2

$A(n,3,2)\leq n(n-1)/3$* for any $n\geq 3$.*

Proof:

Let ${\cal C}$ be an $(n,M,2)$ $2$-signature code of constant-weight $3$ with $M=A(n,3,2)$. According to (9), we have

[TABLE]

since each $3$-elements set has $3$ subsets of size $2$ and each codeword in ${\cal C}$ is calculated three times in the right-hand side of (10). Besides, Lemma IV.2 tells that $\binom{T(A)}{2}\cap\binom{T(A^{\prime})}{2}=\emptyset$ for any distinct $A,A^{\prime}\in\binom{[n]}{2}$, implying

[TABLE]

Note that the maximum value of $\sum_{A\in\binom{[n]}{2}}|T(A)|$ under the condition (11) is achieved when $|T(A)|=2$ for all $A\in\binom{[n]}{2}$. Hence by (10),

[TABLE]

and the theorem follows. ∎

Next, we show that the upper bound on $A(n,3,2)$ in Theorem IV.2 can be achieved for all values of $n$ with some exceptions by establishing the relationship between signature codes and weakly union-free families. A family ${\cal F}\subseteq 2^{[n]}$ is called weakly union-free if for any distinct $A,B,C,D\in{\cal F}$, $A\cup B\neq C\cup D$.

Lemma IV.3

Let ${\cal C}$ be a binary code of length $n$ and ${\cal F}\subseteq 2^{[n]}$ be the corresponding family. If ${\cal F}$ is weakly union-free, then ${\cal C}$ is a $2$-signature code.

Proof:

Assume that ${\cal F}$ is weakly union-free, but ${\cal C}$ is not a $2$-signature code. By Lemma II.1, there exist distinct codewords \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}\in{\cal C} such that \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{s}. This implies that \mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{i})\cup\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j})=\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k})\cup\mathrm{supp}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{l}), a contradiction to the assumption. Then the lemma follows. ∎

We remark that the converse of Lemma IV.3 does not hold. That is, if ${\cal C}$ is a $2$-signature code, the corresponding family ${\cal F}$ may not be weakly union-free. The following is an example.

Example 4

Let ${\cal C}$ be a binary code of length $5$ and constant-weight $3$ defined as below.

[TABLE]

Its corresponding family is ${\cal F}=\{\{1,2,3\},\{3,4,5\},\{1,3,4\},\{2,4,5\}\}$. It is easy to check that ${\cal C}$ is a $2$-signature code, but ${\cal F}$ is not weakly union-free since $\{1,2,3\}\cup\{3,4,5\}=\{1,3,4\}\cup\{2,4,5\}$.

Denote $F_{3}(n)$ as the maximum size of a weakly union-free family ${\cal F}\subseteq\binom{[n]}{3}$ and a weakly union-free family ${\cal F}\subseteq\binom{[n]}{3}$ is called optimal if it has size $F_{3}(n)$. Then by Lemma IV.3, we have $A(n,3,2)\geq F_{3}(n)$. In [15, 8, 27], the exact value of $F_{3}(n)=n(n-1)/3$ for any $n\geq 24$ and $n\in\{13,16,17,19,20,21\}$ was proved and direct constructions from combinatorial design theory for optimal weakly union-free families ${\cal F}\subseteq\binom{[n]}{3}$ were presented. Combining this with Theorem IV.2 and Lemma IV.3, we have

Theorem IV.3

$A(n,3,2)=n(n-1)/3$* for any $n\geq 24$ and $n\in\{13,16,17,19,20,21\}$.*

We remark that by Theorem IV.3, optimal $2$-signature codes of length $n$ and constant-weight $3$ for $n\geq 24$ and $n\in\{13,16,17,19,20,21\}$ can be constructed from optimal weakly union-free families ${\cal F}\subseteq\binom{[n]}{3}$. The interested reader is referred to [15, 8, 27] for more details on the constructions for optimal weakly union-free families ${\cal F}\subseteq\binom{[n]}{3}$.

V Constructions for $t$-signature codes with efficient decoding

In this section, we provide two explicit constructions for signature codes, one is based on concatenation and the other on the Kronecker product. We prove that the signature codes obtained from these two constructions have efficient decoding algorithms and large code sizes. Moreover, we propose the concept of two-level signature code, and show that the product construction could be applied to construct two-level signature codes.

We first recap the definitions of frameproof code and superimposed code which will be exploited later. As stated in Section I, Cheng and Miao [12] proved that frameproof codes have an excellent traceability in a discretized model of multimedia fingerprinting, which promoted the study on frameproof codes, see [46] for example. Superimposed codes were proposed by Kautz and Singleton [36] for retrieving files, and later also extensively investigated in the contexts of disjunct matrices and cover-free families, see [16, 24] for example.

Let ${\cal C}$ be an $(n,M,q)$ code. For any $1\leq i\leq n$, denote {\cal C}(i)=\{c(i):\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}=(c(1),\ldots,c(n))\in{\cal C}\} as the set of the $i$-th coordinates of ${\cal C}$. The descendant code of ${\cal C}$ is defined as

[TABLE]

Definition V.1

Let $t\geq 2$ be an integer.

${\cal C}$* is an $(n,M,q)$ $t$-frameproof code, or an $(n,M,q)$ $t$-FPC, if for any subset ${\cal C}_{0}\subseteq{\cal C}$ with $1\leq|{\cal C}_{0}|\leq t$, we have $\mathrm{desc}({\cal C}_{0})\cap{\cal C}={\cal C}_{0}$, that is, for any other \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}=(c(1),\ldots,c(n))\in{\cal C}\setminus{\cal C}_{0}, there exists $1\leq k\leq n$ such that $c(k)\not\in{\cal C}_{0}(k)$.* 2. 2.

${\cal C}$* is an $(n,M,2)$ $t$-superimposed code, if for any subset ${\cal C}_{0}\subseteq{\cal C}$ with $1\leq|{\cal C}_{0}|\leq t$ and any other codeword $\bm{c}=(c(1),\ldots,c(n))\in{\cal C}\setminus{\cal C}_{0}$, there exists $1\leq k\leq n$ such that $c(k)=1$ and ${\cal C}_{0}(k)=\{0\}$.*

V-A Concatenated construction

We now give the concatenated construction by taking a small $t$-signature code as the inner code which guarantees the concatenated code to be a $t$-signature code, and taking a $q$-ary $t$-FPC as the outer code which makes sure the concatenated code can reduce complexity of decoding compared to an arbitrary $t$-signature code of the same length and size.

Construction 1

Let ${\cal A}$ be an $(n_{1},M_{1},q)$ $t$-FPC over the alphabet $Q$ and ${\cal B}$ be an $(n_{2},q,2)$ $t$-signature code. Define a bijection $\phi:\ Q\rightarrow{\cal B}$. Let ${\cal C}$ be the code defined by

[TABLE]

Then ${\cal C}$ is an $(n_{1}n_{2},M_{1},2)$ $t$-signature code.

Proof:

It is obvious that ${\cal C}$ is an $(n_{1}n_{2},M_{1},2)$ code. For any distinct ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, and for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1, we would like to show

[TABLE]

Since ${\cal C}_{1}\neq{\cal C}_{2}$, without loss of generality, we may assume that there exists one codeword $\bm{c}$ such that \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal C}_{1}\setminus{\cal C}_{2}. Suppose that ${\cal C}_{1},{\cal C}_{2}$ correspond to ${\cal A}_{1},{\cal A}_{2}\subseteq{\cal A}$ respectively, and $\textstyle c$ corresponds to \mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}=(a(1),\ldots,a(n_{1}))\in{\cal A}. Clearly, $|{\cal A}_{1}|\leq t$, $|{\cal A}_{2}|\leq t$ and \mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}\in{\cal A}_{1}\setminus{\cal A}_{2}. Since ${\cal A}$ is an $(n_{1},M_{1},q)$ $t$-FPC, there must exist one coordinate $i$, $1\leq i\leq n_{1}$ such that $a(i)\not\in{\cal A}_{2}(i)$. This implies that ${\cal A}_{1}(i)\neq{\cal A}_{2}(i)$. Let $I_{i}=\{(i-1)n_{2}+1,(i-1)n_{2}+2,\ldots,in_{2}\}$. Then \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}|_{I_{i}}:\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}\} and \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}|_{I_{i}}:\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}\} are two multi-subsets of ${\cal B}$ with distinct base sets. Since the inner code ${\cal B}$ is a $t$-signature code, by Proposition II.1 and the fact that $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, we have

[TABLE]

for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1, which implies (12). Hence ${\cal C}$ is a $t$-signature code. ∎

In the literature, FPCs are usually constructed from structures in combinatorial design theory and coding theory such as orthogonal array, packing designs and error-correcting codes, see [3, 4, 9, 48, 45] for example. As stated in Section II-C, constructions for signature codes are only known in [23] where Goppa codes in coding theory were exploited. The following is an example to describe Construction 1 where we use a $2$-FPC constructed from an orthogonal array in [3] as the outer code and use a $2$-signature code of length $2$ shown in Corollary III.3 as the inner code.

Example 5

Let ${\cal A}$ be a $(2,9,3)$ $2$-FPC and ${\cal B}$ be a $(2,3,2)$ $2$-signature code defined below.

[TABLE]

Define the bijection $\phi$ from $Q=\{0,1,2\}$ to {\cal B}=\{\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}_{1},\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}_{2},\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}_{3}\} as

[TABLE]

By Construction 1, the concatenated code ${\cal C}$ based on the outer code ${\cal A}$ and inner code ${\cal B}$ is a $(6,9,2)$ $2$-signature code presented below.

[TABLE]

We remark that the signature code obtained from Construction 1 has larger length and code size than the inner code which is also a signature code, but may be not good in terms of code rate. For example, the code rate of the concatenated signature code ${\cal C}$ in Example 5 is $(\log_{2}9)/6\approx 0.5283$, while the code rate of the inner code ${\cal B}$ is $(\log_{2}3)/2\approx 0.7925$. However, we will show that the concatenated signature code provides an efficient tracing algorithm which consists of two steps, that is, first decoding the inner code and then decoding the outer code.

Theorem V.1

The concatenated signature code ${\cal C}$ obtained by Construction 1 can trace back to a coalition of size at most $t$ in time $O(n_{1}n_{2}q^{t}+n_{1}M_{1})$.

Proof:

Suppose that \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}=(\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{1},\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{2},\ldots,\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{n_{1}}) is an output of multimedia fingerprinting channel where \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}=(r_{i}(1),\ldots,r_{i}(n_{2})) for all $1\leq i\leq n_{1}$. Suppose that ${\cal X}\subseteq{\cal C}$, $|{\cal X}|\leq t$ corresponds to the real coalition for the output $\textstyle r$. To determine ${\cal X}$ by $\textstyle r$ and the code ${\cal C}$, we divide the decoding process into the following two steps.

Step 1. First we decode the inner code. For each \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}, $1\leq i\leq n_{1}$, a subset ${\cal B}_{i}\subseteq{\cal B}$, $|{\cal B}_{i}|=s_{i}\leq t$ can be traced back since the inner code ${\cal B}$ is a $t$-signature code. Denote Q_{i}=\{\phi^{-1}(\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}):\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}\in{\cal B}_{i}\}\subseteq Q, $1\leq i\leq n_{1}$. The time cost in this step is $O(n_{1}n_{2}q^{t})$.

Step 2. Next we decode the outer code. For the outer code ${\cal A}$, detect each codeword \mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}=(a(1),\ldots,a(n_{1}))\in{\cal A} by checking if $a(i)\in Q_{i}$ for all $1\leq i\leq n_{1}$. If so, then the user corresponding to the codeword \Phi(\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}})\in{\cal C} is identified as a colluder. Denote \widehat{{\cal X}}=\{\Phi(\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}})\in{\cal C}:\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}\in{\cal A}\ \text{and}\ a(i)\in Q_{i},\forall 1\leq i\leq n_{1}\}. The time cost in this step is $O(n_{1}M_{1})$.

Now we show that ${\cal X}$ will be identified after Steps 1 and 2. To this end, we show that ${\cal X}=\widehat{{\cal X}}$. By Step 1, we have ${\cal X}\subseteq\widehat{{\cal X}}$. Assume that there exists \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{0}\in\widehat{{\cal X}}\setminus{\cal X}. Then by Step 2, we have \Phi^{-1}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{0})\in\mathrm{desc}(\{\Phi^{-1}(\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}):\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal X}\}), a contradiction to the condition that ${\cal A}$ is a $t$-FPC. Hence we have ${\cal X}=\widehat{{\cal X}}$.

Based on Steps 1 and 2, the concatenated signature code ${\cal C}$ can trace back to a coalition of size at most $t$ in time $O(n_{1}n_{2}q^{t}+n_{1}M_{1})$. ∎

In general, as described in Section II, the decoding complexity of an $(n_{1}n_{2},M_{1},2)$ $t$-signature code is $O(n_{1}n_{2}M_{1}^{t})$. According to Theorem V.1, the $(n_{1}n_{2},M_{1},2)$ $t$-signature code obtained by Construction 1 can reduce the decoding complexity to $O(n_{1}n_{2}q^{t}+n_{1}M_{1})$ if we choose an $(n_{1},M_{1},q)$ $t$-FPC with $q<M_{1}$ as the outer code. This can be achieved due to the known results on $t$-FPCs, see [4] for example. The following is an example to illustrate the decoding process described in Theorem V.1.

Example 6

Consider the concatenated $(6,9,2)$ $2$-signature code ${\cal C}$ in Example 5 being applied to construct fingerprints for $9$ authorized users in multimedia fingerprinting. Each codeword \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C} corresponds to the fingerprint of user $j$, $1\leq j\leq 9$. Suppose that {\cal X}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{9}\} represents the coalition set and $\textstyle r$ is the output of multimedia fingerprinting channel generated by ${\cal X}$ as

[TABLE]

where \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{1}=(1,0.7)^{\top}, \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{2}=(0.3,0.7)^{\top} and \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{3}=(1,0)^{\top}. We decode ${\cal X}$ by $\textstyle r$ and ${\cal C}$ in the following two steps.

Step 1. As stated in Example 5, the inner code ${\cal B}$ in (13) is a $(2,3,2)$ $2$-signature code. For each \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}, $1\leq i\leq 3$, we can decode that

[TABLE]

where $I_{1}=\{1,2\}$, $I_{2}=\{3,4\}$ and $I_{3}=\{5,6\}$.

Step 2. Denote ${\cal Y}\subseteq{\cal A}$ as the subset of the outer code corresponding to ${\cal X}$. Then by (13) and (14), we obtain

[TABLE]

To determine ${\cal X}$, it is equivalent to determine ${\cal Y}$. As stated in Example 5, the outer code ${\cal A}$ is a $2$-FPC. We can decode {\cal Y}=\{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{1},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{9}\} by choosing each \mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{j}, $1\leq j\leq 9$, such that $a_{j}(i)\in{\cal Y}(i)$ for any $1\leq i\leq 3$.

V-B Product construction

Now we provide a product construction by combining superimposed codes and signature codes as follows. The ingredient $t$-signature code makes the constructed code $t$-signature, and the ingredient $t$-superimposed code allows for efficient decoding. Our constructed code can reduce decoding complexity compared to an arbitrary $t$-signature code of the same length and size.

Construction 2

Let ${\cal A}$ be an $(n_{1},M_{1},2)$ $t$-superimposed code, and ${\cal B}$ be an $(n_{2},M_{2},2)$ $t$-signature code and \mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\in{\cal B}. Denote {\cal B}^{\ast}={\cal B}\setminus\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\}. Let ${\cal C}={\cal A}\otimes{\cal B}^{\ast}$, where ${\cal A}\otimes{\cal B}^{\ast}$ is the Kronecker product of ${\cal A}$ and ${\cal B}^{\ast}$:

[TABLE]

Then ${\cal C}$ is an $(n_{1}n_{2},M_{1}(M_{2}-1),2)$ $t$-signature code.

Proof:

It is obvious that ${\cal B}^{\ast}$ is an $(n_{2},M_{2}-1,2)$ code, and thus ${\cal C}$ is an $(n_{1}n_{2},M_{1}(M_{2}-1),2)$ code. For simplicity, we divide ${\cal C}$ into $M_{1}$ groups of codewords as

[TABLE]

where

[TABLE]

For any two distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ such that $|{\cal C}_{1}|\leq t$ and $|{\cal C}_{2}|\leq t$, denote $G_{1}=\{h\in[M_{1}]:{\cal C}_{1}\cap{\cal G}_{h}\neq\emptyset\}$ and $G_{2}=\{h\in[M_{1}]:{\cal C}_{2}\cap{\cal G}_{h}\neq\emptyset\}$. Obviously, $|G_{1}|\leq t$ and $|G_{2}|\leq t$. We would like to show that for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1, we have

[TABLE]

To this end, we consider the following two cases.

$G_{1}=G_{2}$ and $|G_{1}|=|G_{2}|=s$, $1\leq s\leq t$.

1. 1.a)

If $s=1$, without loss of generality, we may assume $G_{1}=G_{2}=\{1\}$. Then ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal G}_{1}$. Since ${\cal A}$ is an $(n_{1},M_{1},2)$ $t$-superimposed code, ${\cal A}$ does not have $\textstyle 0$ as a column vector. Then there must exist $i$, $1\leq i\leq n_{1}$ such that $a_{1}(i)=1$. Denote $I_{i}=\{(i-1)n_{2}+1,(i-1)n_{2}+2,\ldots,in_{2}\}$, $1\leq i\leq n_{1}$. Recall that ${\cal C}_{1}\neq{\cal C}_{2}$. Then, according to (15), \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}|_{I_{i}}:\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}\} and \{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}|_{I_{i}}:\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}\} are two distinct subsets of ${\cal B}^{\ast}$. Since ${\cal B}^{\ast}$ is a $t$-signature code, we have

[TABLE]

for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1, implying (16). 2. 1.b)

If $2\leq s\leq t$, without loss of generality, we may assume $G_{1}=G_{2}=\{1,\ldots,s\}$. Then $|{\cal C}_{1}\cap{\cal G}_{h}|\leq t-1$ and $|{\cal C}_{2}\cap{\cal G}_{h}|\leq t-1$ for any $h\in G_{1}$. Since ${\cal C}_{1}\neq{\cal C}_{2}$, there exists at least one $h\in G_{1}$ such that ${\cal C}_{1}\cap{\cal G}_{h}\neq{\cal C}_{2}\cap{\cal G}_{h}$. Without loss of generality, we may assume ${\cal C}_{1}\cap{\cal G}_{1}\neq{\cal C}_{2}\cap{\cal G}_{1}$. Since ${\cal A}$ is a $t$-superimposed code, there exists $i$, $1\leq i\leq n_{1}$ such that $a_{1}(i)=1$ and $a_{h}(i)=0$ for any $2\leq h\leq s$. Clearly, \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}|_{I_{i}}=\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}} for any \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}\in{\cal G}_{h}, $2\leq h\leq s$. Since {\cal B}={\cal B}^{\ast}\cup\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\} is a $t$-signature code, we have

[TABLE]

for any real numbers $\lambda_{j},\lambda,\lambda_{k}^{\prime},\lambda^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}\cap{\cal G}_{1}}\lambda_{j}+\lambda=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}\cap{\cal G}_{1}}\lambda_{k}^{\prime}+\lambda^{\prime}=1. This implies that

[TABLE]

for any real numbers $\lambda_{j},\lambda_{m},\lambda_{k}^{\prime},\lambda_{r}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}\cap{\cal G}_{1}}\lambda_{j}+\sum_{h=2}^{s}\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{m}\in{\cal C}_{1}\cap{\cal G}_{h}}\lambda_{m}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}\cap{\cal G}_{1}}\lambda_{k}^{\prime}+\sum_{h=2}^{s}\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{r}\in{\cal C}_{2}\cap{\cal G}_{h}}\lambda_{r}^{\prime}=1, which further implies (16). 2. 2.

$G_{1}\neq G_{2}$. Without loss of generality, we may assume $G_{1}\setminus G_{2}\neq\emptyset$ and $h\in G_{1}\setminus G_{2}$. Since ${\cal A}$ is a $t$-superimposed code, there exists $i$, $1\leq i\leq n_{1}$ such that $a_{h}(i)=1$ and $a_{h^{\prime}}(i)=0$ for all $h^{\prime}\in G_{2}$. By the construction of ${\cal C}$ and the fact that {\cal B}^{\ast}={\cal B}\setminus\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\}, ${\cal C}_{1}|_{I_{i}}$ contains at least one nonzero vector and ${\cal C}_{2}|_{I_{i}}=\{\bm{0}\}$, which implies that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}|_{I_{i}}\neq\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}} and \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}|_{I_{i}}=\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}} for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1. Then (16) follows.

Hence ${\cal C}$ is a $t$-signature code. ∎

In the literature, constructions for superimposed codes were widely investigated. In [36], Kautz and Singleton constructed superimposed codes from error-correcting codes such as maximum distance separable codes in coding theory. Macula [40] proposed a way of constructing disjunct matrices from a combinatorial viewpoint where the containment relation among sets was used. Combinatorial structures in combinatorial design theory such as packing designs and orthogonal arrays were used to construct cover-free families, see [24, 36, 48] for example. In what follows, we use an example to illustrate Construction 2 in which the $3\times 3$ identity matrix, which is a $(3,3,2)$ $2$-superimposed code by Definition V.1, and the $(3,5,2)$ $2$-signature code shown in Example 2 are employed.

Example 7

Let ${\cal A}$ be a $(3,3,2)$ $2$-superimposed code and ${\cal B}$ be a $(3,5,2)$ $2$-signature code defined below.

[TABLE]

Then by Construction 2, ${\cal C}={\cal A}\otimes{\cal B}^{\ast}$ is a $(9,12,2)$ $2$-signature code presented below.

[TABLE]

We remark that the signature code ${\cal C}$ obtained from Construction 2 has larger length and code size than the ingredient signature code ${\cal B}$, but may be not good in the sense of code rate. For example, the code rate of the signature code ${\cal C}$ in Example 7 is $(\log_{2}12)/9\approx 0.3983$, while the code rate of the ingredient signature code ${\cal B}$ is $(\log_{2}5)/3\approx 0.7740$. However, we will show that the signature code obtained from Construction 2 provides an efficient tracing algorithm which consists of two steps, that is, first decoding the ingredient superimposed code and then decoding the ingredient signature code. Notice that this is in some sense the converse order of the decoding in Theorem V.1.

Theorem V.2

The signature code ${\cal C}$ obtained by Construction 2 can trace back to a coalition of size at most $t$ in time $O(tn_{2}M_{2}^{t}+n_{1}n_{2}M_{1}+tn_{1})$.

Proof:

Suppose that \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}=(\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{1},\ldots,\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{n_{1}}) is an output of the multimedia fingerprinting channel where \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}=(r_{i}(1),\ldots,r_{i}(n_{2})) for all $1\leq i\leq n_{1}$. Suppose that ${\cal X}\subseteq{\cal C}$, $|{\cal X}|\leq t$ corresponds to the real coalition for the output $\textstyle r$. Denote $G=\{h\in[M_{1}]:{\cal X}\cap{\cal G}_{h}\neq\emptyset\}$ as the set of group indices of the codewords in ${\cal X}$. Obviously, $|G|\leq t$. To determine ${\cal X}$ by $\textstyle r$ and the code ${\cal C}$, we will first determine $G$, and then determine ${\cal X}\cap{\cal G}_{h}$ for any $h\in G$.

Step 1. For the ingredient superimposed code ${\cal A}$, detect each codeword $(a_{h}(1),\ldots,a_{h}(n_{1}))\in{\cal A}$, $1\leq h\leq M_{1}$ by checking if there exists $i$, $1\leq i\leq n_{1}$ such that \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}=\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}} and $a_{h}(i)=1$. Denote G_{0}=\{h\in[M_{1}]:\exists 1\leq i\leq n_{1}\ \text{s.t.}\ \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}=\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\ \text{and}\ a_{h}(i)=1\} and $\widehat{G}=[M_{1}]\setminus G_{0}$. We claim that $G=\widehat{G}$. The time cost in this step is $O(n_{1}n_{2}M_{1})$.

To verify our claim, we first show $G\subseteq\widehat{G}$. Assume that there exists $h\in G$ but $h\not\in\widehat{G}$, that is, $h\in G_{0}$. Then there exists $i$, $1\leq i\leq n_{1}$ such that \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}=\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}} and $a_{h}(i)=1$. Notice that \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}=\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}} implies {\cal X}|_{I_{i}}=\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\}. Then we have ${\cal X}\cap{\cal G}_{h}=\emptyset$, that is, $h\not\in G$, a contradiction to the assumption. So we have $G\subseteq\widehat{G}$. On the other hand, we show $\widehat{G}\subseteq G$. Assume that there exists $h^{\prime}\in\widehat{G}$ but $h^{\prime}\not\in G$. Since ${\cal A}$ is a $t$-superimposed code, there exists $i^{\prime}$, $1\leq i^{\prime}\leq n_{1}$ such that $a_{h^{\prime}}(i^{\prime})=1$ and $a_{h}(i^{\prime})=0$ for any $h\in G$. Recall that $G$ is the set of group indices of the codewords in ${\cal X}$. Thus we have {\cal X}|_{I_{i^{\prime}}}=\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\} which implies \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i^{\prime}}=\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}. Then we have $h^{\prime}\in G_{0}$, a contradiction to the assumption that $h^{\prime}\in\widehat{G}$ since $\widehat{G}\cap G_{0}=\emptyset$. So, we have $G=\widehat{G}$ and thus $G$ can be determined after Step 1.

Step 2. In this step, we show how to determine ${\cal X}\cap{\cal G}_{h}$ for any $h\in G$.

If $|G|=1$, without loss of generality, we may assume $G=\{1\}$. Then ${\cal X}={\cal X}\cap{\cal G}_{1}$. Since ${\cal A}$ is a $t$-superimposed code, there exists $i$, $1\leq i\leq n_{1}$ such that $a_{1}(i)=1$. Then we have \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}\neq\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}. Since ${\cal B}^{\ast}$ is a $t$-signature code, we can determine ${\cal X}|_{I_{i}}$ by \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}, and thus can determine ${\cal X}$. The time cost in this case is $O(n_{2}(M_{2}-1)^{t}+n_{1})$. 2. 2.

If $|G|=s\geq 2$, then we have $2\leq\sum_{h\in G}|{\cal X}\cap{\cal G}_{h}|\leq t$, and $1\leq|{\cal X}\cap{\cal G}_{h}|\leq t-1$ for any $h\in G$. Since ${\cal A}$ is a $t$-superimposed code, for any $h\in G$, there exists $i$, $1\leq i\leq n_{1}$ such that $a_{h}(i)=1$ and $a_{h^{\prime}}(i)=0$ for any $h^{\prime}\in G\setminus\{h\}$. Then we have ({\cal X}\setminus{\cal G}_{h})|_{I_{i}}=\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\}. Since {\cal B}={\cal B}^{\ast}\cup\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\} is a $t$-signature code, we can determine $({\cal X}\cap{\cal G}_{h})|_{I_{i}}$ by \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}, and thus can determine ${\cal X}\cap{\cal G}_{h}$ for any $h\in G$. The time cost in this case is $O(tn_{2}M_{2}^{t}+tn_{1})$.

Based on Steps 1 and 2, the $t$-signature code obtained by Construction 2 can trace back to all the colluders of size no more than $t$ in time $O(tn_{2}M_{2}^{t}+n_{1}n_{2}M_{1}+tn_{1})$. ∎

In general, the decoding complexity of an $(n_{1}n_{2},M_{1}(M_{2}-1),2)$ $t$-signature code is $O(n_{1}n_{2}M_{1}^{t}(M_{2}-1)^{t})$. Here according to Theorem V.2, we can reduce the decoding complexity to $O(tn_{2}M_{2}^{t}+n_{1}n_{2}M_{1}+tn_{1})$ by using the $t$-signature code obtained from Construction 2. The following is an example to show the decoding process described in Theorem V.2.

Example 8

Consider the $(9,12,2)$ $2$-signature code ${\cal C}$ in Example 7 being applied to construct fingerprints for $12$ authorized users in multimedia fingerprinting. Each codeword \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C} corresponds to the fingerprint of user $j$, $1\leq j\leq 12$. We divide all the codewords of ${\cal C}$ into $3$ groups ${\cal C}={\cal G}_{1}\cup{\cal G}_{2}\cup{\cal G}_{3}$ where {\cal G}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{1},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}\}, {\cal G}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{5},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{8}\} and {\cal G}_{3}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{9},\ldots,\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{12}\}, and divide all the coordinates of the codewords in ${\cal C}$ into $I_{1}=\{1,2,3\}$, $I_{2}=\{4,5,6\}$ and $I_{3}=\{7,8,9\}$. Suppose that ${\cal X}\subseteq{\cal C}$ with $|{\cal X}|\leq 2$ represents the coalition set, $\textstyle r$ is the output of multimedia fingerprinting channel generated by ${\cal X}$, and $G=\{1\leq h\leq 3:{\cal X}\cap{\cal G}_{h}\neq\emptyset\}$ is the set of group indices of the codewords in ${\cal X}$. We discuss two cases on the distributions of codewords in ${\cal X}$.

Suppose that {\cal X}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}\} and the output $\textstyle r$ generated by ${\cal X}$ is

[TABLE]

where \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{1}=(0.6,1,0.4)^{\top} and \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{2}=\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{3}=(0,0,0)^{\top}. We decode ${\cal X}$ by $\textstyle r$ and ${\cal C}$ in the following two steps.

Step 1. As shown in (17) in Example 7, ${\cal A}$ is a $2$-superimposed code and $a_{2}(2)=a_{3}(3)=1$. Since \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{2}=\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{3}=(0,0,0)^{\top}, we can decode $G=\{1\}$, implying that ${\cal X}\subseteq{\cal G}_{1}$.

Step 2. Based on Step 1, we focus on ${\cal G}_{1}$. As shown in (17), ${\cal B}^{\ast}$ is a $2$-signature code, then we can decode {\cal X}|_{I_{1}}=\{\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}_{3},\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}_{5}\} by \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{1}, which implies {\cal X}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}\}. 2. 2)

Suppose that {\cal X}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{7}\} and the output $\textstyle r$ generated by ${\cal X}$ is

[TABLE]

where \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{1}=(0,0.5,0.5)^{\top}, \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{2}=(0.5,0,0.5)^{\top} and \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{3}=(0,0,0)^{\top}. We decode ${\cal X}$ by $\textstyle r$ and ${\cal C}$ in the following two steps.

Step 1. By \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{3}=(0,0,0)^{\top} and $a_{3}(3)=1$, we first can decode $G=\{1,2\}$, implying that ${\cal X}=({\cal X}\cap{\cal G}_{1})\cup({\cal X}\cap{\cal G}_{2})$.

Step 2. In this step, we determine ${\cal X}\cap{\cal G}_{1}$ and ${\cal X}\cap{\cal G}_{2}$. Note that {\cal B}={\cal B}^{\ast}\cup\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\} is a $2$-signature code. Then by \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{1}, we can determine {\cal X}|_{I_{1}}=\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}},\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}_{3}\}. Since $a_{2}(1)=0$, we have ({\cal X}\cap{\cal G}_{2})|_{I_{1}}=\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\}, implying ({\cal X}\cap{\cal G}_{1})|_{I_{1}}=\{\mathchoice{\mbox{\boldmath\displaystyle b}}{\mbox{\boldmath\textstyle b}}{\mbox{\boldmath\scriptstyle b}}{\mbox{\boldmath\scriptscriptstyle b}}_{3}\}. From the construction of ${\cal C}$ in Example 7, we obtain {\cal X}\cap{\cal G}_{1}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2}\}. Similarly, we can decode {\cal X}\cap{\cal G}_{2}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{7}\} by \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{2}. Thus, we have {\cal X}=\{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{2},\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{7}\}.

V-C Two-level signature code

In this subsection, we show that Construction 2 can be applied to construct two-level signature codes as well. Two-level fingerprinting codes were first investigated by Anthapadmanabhan and Barg [2] in digital fingerprinting for the purpose of getting partial information if the size of a coalition exceeds a predetermined threshold. As shown in [2], users accommodated in the digital fingerprinting system were partitioned into several groups and two-level $(t_{1},t_{2})$-traceability codes with $t_{1}>t_{2}$ were introduced to guarantee that once a pirate copy produced by a coalition is confiscated, the dealer can: 1) identify at least one colluder if the coalition size is no more than $t_{2}$, which is the same to a traditional $t_{2}$-traceability code (one-level), and 2) trace back to at least one group that contains at least one colluder if the coalition size is larger than $t_{2}$ but no more than $t_{1}$. Inspired by the idea and the applications of two-level traceability codes, in the literature, several other types of two-level fingerprinting codes were proposed, see [1, 43] for example. Here, we introduce the concept of two-level signature code for multimedia fingerprinting. To the best of our knowledge, this is the first work to introduce this concept.

Definition V.2

Let ${\cal C}$ be an $(n,M,2)$ code and $t_{1},t_{2}$ be positive integers with $t_{1}>t_{2}$. Suppose that all the codewords of ${\cal C}$ are partitioned into $M_{1}(<M)$ groups: ${\cal C}={\cal G}_{1}\cup{\cal G}_{2}\cup\ldots\cup{\cal G}_{M_{1}}$. For any subset ${\cal C}^{\prime}\subseteq{\cal C}$, denote $G({\cal C}^{\prime})=\{h\in[M_{1}]:{\cal C}^{\prime}\cap{\cal G}_{h}\neq\emptyset\}$ as the set of group indices of the codewords in ${\cal C}^{\prime}$. ${\cal C}$ is an $(n,M,2)$ two-level $(t_{1},t_{2})$-signature code if for any two subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ and for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1,

[TABLE]

implies that

${\cal C}_{1}={\cal C}_{2}$* if $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t_{2}$;* 2. 2.

$G({\cal C}_{1})=G({\cal C}_{2})$* if $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t_{1}$.*

By Definition V.2, in multimedia fingerprinting, a two-level $(t_{1},t_{2})$-signature code could

1. a)

trace back to all the colluders if the coalition size is at most $t_{2}$; 2. b)

determine all the groups each of which contains at least one colluder if the coalition size is at most $t_{1}$.

We have the following relationship between signature codes (one-level) and two-level signature codes.

Lemma V.1

Let $t_{1},t_{2}$ be two positive integers such that $t_{1}>t_{2}$. Then a $t_{1}$-signature code is a two-level $(t_{1},t_{2})$-signature code, and a two-level $(t_{1},t_{2})$-signature code is a $t_{2}$-signature code.

Proof:

By Definitions II.1 and V.2, it is obvious that a two-level $(t_{1},t_{2})$-signature code is a $t_{2}$-signature code.

Suppose that ${\cal C}$ is a $t_{1}$-signature code, then we can partition all the codewords of ${\cal C}$ into non-empty groups.

By Definition II.1 and the condition that $t_{1}>t_{2}$, ${\cal C}$ is a $t_{2}$-signature code; 2. 2.

For any two subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$, if $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t_{1}$ and the group indices of ${\cal C}_{1},{\cal C}_{2}$ are different, then ${\cal C}_{1}\neq{\cal C}_{2}$. By Definition II.1, for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1, we have \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\neq\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}.

Thus, by Definition V.2, ${\cal C}$ is a $(t_{1},t_{2})$-signature code. The lemma follows. ∎

Lemma V.1 implies that a $(t_{1},t_{2})$-signature code lies between a $t_{1}$-signature code and a $t_{2}$-signature code. The following construction for two-level signature codes is an application of Construction 2.

Theorem V.3

Let $t_{1},t_{2}$ be two positive integers such that $t_{1}>t_{2}$. Let ${\cal A}$ be an $(n_{1},M_{1},2)$ $t_{1}$-superimposed code and ${\cal B}$ be an $(n_{2},M_{2},2)$ $t_{2}$-signature code containing $\textstyle 0$. Let {\cal B}^{\ast}={\cal B}\setminus\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\}. Then ${\cal C}={\cal A}\otimes{\cal B}^{\ast}$ defined by (15) is an $(n_{1}n_{2},M_{1}(M_{2}-1),2)$ two-level $(t_{1},t_{2})$-signature code with $M_{1}$ groups.

Proof:

By Construction 2, an $(n_{1}n_{2},M_{1}(M_{2}-1),2)$ code ${\cal C}$ can be obtained from ${\cal A}$ and ${\cal B}$. Now we show that ${\cal C}$ is a two-level $(t_{1},t_{2})$-signature code. It is obvious that all the codewords of ${\cal C}$ are divided into $M_{1}$ groups and each group contains $M_{2}-1$ codewords. If a forged copy is created by a coalition of size at most $t_{2}$, all the colluders will be identified by Theorem V.2. If a forged copy is created by a coalition of size at most $t_{1}$, by Step 1 of the decoding process in the argument of Theorem V.2, any group containing at least one colluder will be identified. The proof is completed. ∎

VI Comments on noisy weighted binary adder channel and multimedia fingerprinting

In the previous sections, we investigated signature codes for noiseless multimedia fingerprinting channel. In this section, we consider the noisy scenario. First we show from a geometric viewpoint that there does not exist any binary code with complete traceability for noisy multimedia fingerprinting channel. Then we introduce frameproof signature codes for noisy multimedia fingerprinting to protect innocent groups of users by guaranteeing that disjoint coalition sets cannot produce an identical forged copy.

VI-A Noisy multimedia fingerprinting

In practice, the noisy multimedia fingerprinting channel is more realistic but with more complicated assumptions than the noiseless case. That is, the dealer observes the forged copy with some noise which may be produced artificially by the coalition before redistributing the forged copy, or generated unartificially during the redistribution process of the forged copy. We show from a geometric viewpoint that no coalition could be completely traced back in noisy multimedia fingerprinting.

Assume that the colluders in a coalition $J$ add some noise $\textstyle\varepsilon$ for the purpose of making themselves less likely to be identified, where \mathchoice{\mbox{\boldmath\displaystyle\varepsilon}}{\mbox{\boldmath\textstyle\varepsilon}}{\mbox{\boldmath\scriptstyle\varepsilon}}{\mbox{\boldmath\scriptscriptstyle\varepsilon}}=(\varepsilon(1),\ldots,\varepsilon(m))\in\mathbb{R}^{m}\setminus\{\mathchoice{\mbox{\boldmath\displaystyle 0}}{\mbox{\boldmath\textstyle 0}}{\mbox{\boldmath\scriptstyle 0}}{\mbox{\boldmath\scriptscriptstyle 0}}\}. Typically, we consider the adversarial noise with bounded energy, that is, \|\mathchoice{\mbox{\boldmath\displaystyle\varepsilon}}{\mbox{\boldmath\textstyle\varepsilon}}{\mbox{\boldmath\scriptstyle\varepsilon}}{\mbox{\boldmath\scriptscriptstyle\varepsilon}}\|<\delta for some real number $\delta>0$. Then the dealer observes the forged copy

[TABLE]

Notice that $\textstyle\varepsilon$ is chosen by the coalition $J$ and unknown to the dealer. However, the dealer can calculate

[TABLE]

for $1\leq k\leq n$, and obtain

[TABLE]

where \mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}=(e(1),\ldots,e(n))\in\mathbb{R}^{n} and e(k)=\langle\mathchoice{\mbox{\boldmath\displaystyle\varepsilon}}{\mbox{\boldmath\textstyle\varepsilon}}{\mbox{\boldmath\scriptstyle\varepsilon}}{\mbox{\boldmath\scriptscriptstyle\varepsilon}},\mathchoice{\mbox{\boldmath\displaystyle f}}{\mbox{\boldmath\textstyle f}}{\mbox{\boldmath\scriptstyle f}}{\mbox{\boldmath\scriptscriptstyle f}}_{k}\rangle for $1\leq k\leq n$. It should be noted that the dealer does not know what $\textstyle e$ is, but knows that \|\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}\|\leq\|\mathchoice{\mbox{\boldmath\displaystyle\varepsilon}}{\mbox{\boldmath\textstyle\varepsilon}}{\mbox{\boldmath\scriptstyle\varepsilon}}{\mbox{\boldmath\scriptscriptstyle\varepsilon}}\|<\delta.

In noisy multimedia fingerprinting, the dealer would also like to design a binary code ${\cal C}$ with some properties to identify the whole coalition set $J$ based on the result \tilde{\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}} calculated in (20). We define the complete traceability of a binary code for noisy multimedia fingerprinting as follows.

Definition VI.1

Let ${\cal C}$ be an $(n,M,2)$ code, $t\geq 2$ be an integer and $\delta>0$ be a real number. ${\cal C}$ has $(t,\delta)$-complete traceability if for any two distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, we have

[TABLE]

for any \mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}},\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}^{\prime}\in\mathbb{R}^{n} and any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \|\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}\|,\|\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}^{\prime}\|<\delta and \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1.

Let \mathchoice{\mbox{\boldmath\displaystyle z}}{\mbox{\boldmath\textstyle z}}{\mbox{\boldmath\scriptstyle z}}{\mbox{\boldmath\scriptscriptstyle z}}\in\mathbb{R}^{n} be a point in the $n$-dimensional Euclidean space and $\delta>0$ be a real number. An open $n$-ball with center $\textstyle z$ and radius $\delta$ is formed by all the points in \{\mathchoice{\mbox{\boldmath\displaystyle z}}{\mbox{\boldmath\textstyle z}}{\mbox{\boldmath\scriptstyle z}}{\mbox{\boldmath\scriptscriptstyle z}}^{\prime}\in\mathbb{R}^{n}:\|\mathchoice{\mbox{\boldmath\displaystyle z}}{\mbox{\boldmath\textstyle z}}{\mbox{\boldmath\scriptstyle z}}{\mbox{\boldmath\scriptscriptstyle z}}-\mathchoice{\mbox{\boldmath\displaystyle z}}{\mbox{\boldmath\textstyle z}}{\mbox{\boldmath\scriptstyle z}}{\mbox{\boldmath\scriptscriptstyle z}}^{\prime}\|<\delta\}. For any distinct $S_{1},S_{2}\subseteq\mathbb{R}^{n}$, define the distance between $S_{1}$ and $S_{2}$ as

[TABLE]

Note that for any distinct $S_{1},S_{2}\subseteq\mathbb{R}^{n}$, if $S_{1}\cap S_{2}\neq\emptyset$, then we must have $\mathrm{d}(S_{1},S_{2})=0$. We have the following equivalent description for Definition VI.1.

Proposition VI.1

An $(n,M,2)$ code ${\cal C}$ has $(t,\delta)$-complete traceability if and only if for any two distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, we have

[TABLE]

Proof:

Let ${\cal C}$ be an $(n,M,2)$ code, ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ be two distinct subsets with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$ and $\lambda_{j},\lambda_{k}^{\prime}>0$ be real numbers such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1. Then \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j} is a point in $\mathcal{P}({\cal C}_{1})$ and \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k} is a point in $\mathcal{P}({\cal C}_{2})$. Moreover, for any \mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}},\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}^{\prime}\in\mathbb{R}^{n} with \|\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}\|,\|\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}^{\prime}\|<\delta, \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}+\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}} is a point in the open $n$-ball with center \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j} and radius $\delta$, and \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}+\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}^{\prime} is a point in the open $n$-ball with center \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k} and radius $\delta$. Then the argument

[TABLE]

holds if and only if

[TABLE]

Thus, by Definition VI.1, ${\cal C}$ has $(t,\delta)$-complete traceability if and only if for any two distinct subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, (23) holds for any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1, that is, $\mathrm{d}(\mathcal{P}({\cal C}_{1}),\mathcal{P}({\cal C}_{2}))\geq 2\delta$ according to (6). The conclusion follows.

∎

By Proposition VI.1, in noisy multimedia fingerprinting, a coalition with size no more than $t$ can be completely traced back if and only if there exists a binary code ${\cal C}$ such that for any distinct ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$, the condition (22) is satisfied. However, if ${\cal C}_{1}\cap{\cal C}_{2}\neq\emptyset$, we always have $\mathrm{d}(\mathcal{P}({\cal C}_{1}),\mathcal{P}({\cal C}_{2}))=0$, and thus (22) does not hold. For example, in Example 2, let ${\cal C}_{1}=\{v_{1},v_{2},v_{6}\}$ and ${\cal C}_{2}=\{v_{2},v_{3}\}$. Then ${\cal C}_{1}\cap{\cal C}_{2}\neq\emptyset$ and $\mathcal{P}({\cal C}_{1})\cap\mathcal{P}({\cal C}_{2})=\emptyset$, but $\mathrm{d}(\mathcal{P}({\cal C}_{1}),\mathcal{P}({\cal C}_{2}))=0$ according to (21). This immediately implies

Proposition VI.2

There does not exist any binary code with complete traceability in noisy multimedia fingerprinting.

VI-B Frameproof signature code

In the previous subsection, we showed from a geometric viewpoint that there exists no binary code that can trace back to all the colluders in noisy multimedia fingerprinting. In this subsection, we define a binary code with a weaker requirement than that in Definition VI.1, called a frameproof signature code, to provide some security in noisy multimedia fingerprinting. In the literature, Stinson et al. [47] introduced secure frameproof codes in digital fingerprinting to make sure that any illegal copy cannot be generated simultaneously by two disjoint coalition sets. Inspired by this, we introduce the definition of frameproof signature code as follows.

Definition VI.2

Let ${\cal C}$ be an $(n,M,2)$ code, $t\geq 2$ be an integer and $\delta>0$ be a real number. ${\cal C}$ is an $(n,M,2)$ $(t,\delta)$-frameproof signature code if for any two subsets ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$ and ${\cal C}_{1}\cap{\cal C}_{2}=\emptyset$, we have

[TABLE]

for any \mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}},\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}^{\prime}\in\mathbb{R}^{n} and any real numbers $\lambda_{j},\lambda_{k}^{\prime}>0$ such that \|\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}\|,\|\mathchoice{\mbox{\boldmath\displaystyle e}}{\mbox{\boldmath\textstyle e}}{\mbox{\boldmath\scriptstyle e}}{\mbox{\boldmath\scriptscriptstyle e}}^{\prime}\|<\delta and \sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}\in{\cal C}_{1}}\lambda_{j}=\sum_{\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{k}\in{\cal C}_{2}}\lambda_{k}^{\prime}=1.

According to Definitions VI.1 and VI.2, we immediately have a relationship between the frameproof signature code and the code with complete traceability as follows.

Proposition VI.3

Let ${\cal C}$ be an $(n,M,2)$ code, $t\geq 2$ be an integer and $\delta>0$ be a real number. If ${\cal C}$ has $(t,\delta)$-complete traceability, then ${\cal C}$ is a $(t,\delta)$-frameproof signature code, but not vice versa.

We provide an equivalent description of frameproof signature codes from a geometric viewpoint.

Proposition VI.4

${\cal C}$* is an $(n,M,2)$ $(t,\delta)$-frameproof signature code if and only if for any ${\cal C}_{1},{\cal C}_{2}\subseteq{\cal C}$ with $1\leq|{\cal C}_{1}|,|{\cal C}_{2}|\leq t$ and ${\cal C}_{1}\cap{\cal C}_{2}=\emptyset$, we have $\mathrm{d}(\mathcal{P}({\cal C}_{1}),\mathcal{P}({\cal C}_{2}))\geq 2\delta$.*

Example 9

It is easy to see from Example 2 that ${\cal C}=\{(0,0,0),(1,1,0),(1,0,1),(0,1,1)\}$ is a $(2,1/2)$-frameproof signature code.

We remark that although a $(t,\delta)$-frameproof signature code ${\cal C}$ cannot identify any coalition set in noisy multimedia fingerprinting, it can guarantee at least two things:

If ${\cal C}_{1}\subseteq{\cal C}$ is a coalition of size at most $t$, then ${\cal C}_{1}$ cannot frame any ${\cal C}_{2}\subseteq{\cal C}$ with $|{\cal C}_{2}|\leq t$ and ${\cal C}_{1}\cap{\cal C}_{2}=\emptyset$ since they cannot create the same forged copy \tilde{\mathchoice{\mbox{\boldmath\displaystyle y}}{\mbox{\boldmath\textstyle y}}{\mbox{\boldmath\scriptstyle y}}{\mbox{\boldmath\scriptscriptstyle y}}}. 2. 2.

If ${\cal C}_{1}\subseteq{\cal C}$ is a coalition of size at most $t$ and \tilde{\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}} is the corresponding output of noisy multimedia fingerprinting channel, then any ${\cal C}_{2}\subseteq{\cal C}$ with $|{\cal C}_{2}|\leq t$ and \mathrm{d}(\{\tilde{\mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}}\},\mathcal{P}({\cal C}_{2}))<\delta contains at least one colluder.

VII Conclusion

In this paper, we investigated signature codes for the weighted binary adder channel and collusion-resistant multimedia fingerprinting. We showed the relationships between signature codes and other known combinatorial structures and obtained general and asymptotic upper bounds of $t$-signature codes. We explored the combinatorial properties and derived bounds for $2$-signature codes of constant-weights $2$ and $3$, respectively. Moreover, we provided two explicit constructions for $t$-signature codes which have efficient tracing algorithms. We also introduced two-level signature codes and gave an explicit construction for two-level signature codes. At last, we showed from a geometric viewpoint that there does not exist any binary code with complete traceability for noisy multimedia fingerprinting. As a weaker requirement, to make sure that disjoint coalition sets cannot produce an identical forged copy, we introduced frameproof signature codes for noisy multimedia fingerprinting.

It would be of interest to further improve the bounds for signature codes shown in this paper and find more explicit constructions for signature codes. It would also be of interest to investigate the new type of signature codes introduced in this paper for noisy multimedia fingerprinting.

Appendix

Proof:

Assume that ${\cal C}\subseteq\{0,1\}^{4}$ is a $2$-signature code with $|{\cal C}|=8$. Let

[TABLE]

By Lemma II.1, it is easy to check that (24) is still a $2$-signature code by exchanging any two rows or any two columns. Moreover, \{\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}_{1},\ldots,\mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}_{8}\} is also a $2$-signature code where \mathchoice{\mbox{\boldmath\displaystyle x}}{\mbox{\boldmath\textstyle x}}{\mbox{\boldmath\scriptstyle x}}{\mbox{\boldmath\scriptscriptstyle x}}_{i}=(x_{i}(1),x_{i}(2),x_{i}(3),x_{i}(4)) and $x_{i}(j)=1-c_{i}(j)$ for all $1\leq i\leq 8$ and $1\leq j\leq 4$.

First we divide ${\cal C}$ into groups with respect to the first and second rows of (24). Let

[TABLE]

If $|{\cal C}_{0}|\geq 6$, then ${\cal C}_{0}|_{I}$ is a $2$-signature code of length $3$ and $|{\cal C}_{0}|_{I}|=|{\cal C}_{0}|\geq 6$, which contradicts Corollary III.3 that $A(3,2)=5$. Thus, $|{\cal C}_{0}|\leq 5$. Similarly, we also have $|{\cal C}_{1}|\leq 5$. Since ${\cal C}={\cal C}_{0}\cup{\cal C}_{1}$ and $|{\cal C}|=8$, we only need to discuss the following two cases.

$|{\cal C}_{0}|=3$ and $|{\cal C}_{1}|=5$. Then $|{\cal Y}_{0}|+|{\cal Y}_{1}|=5$ and $|{\cal T}_{0}|+|{\cal T}_{1}|=3$. By Corollary III.3 that $A(2,2)=3$, we have $|{\cal Y}_{0}|\leq 3$ and $|{\cal Y}_{1}|\leq 3$. There are two subcases.

1. 1.1)

$|{\cal Y}_{0}|=2$ and $|{\cal Y}_{1}|=3$. Without loss of generality, we may assume that

[TABLE]

• 1.1.a)

If $|{\cal T}_{0}|=3$, then $|{\cal T}_{0}|_{I^{\prime}}|=3$. Since $|{\cal Y}_{1}|_{I^{\prime}}|=3$ and ${\cal Y}_{1}|_{I^{\prime}},{\cal T}_{0}|_{I^{\prime}}\subseteq\{0,1\}^{2}$, we have $|{\cal Y}_{1}|_{I^{\prime}}\cap{\cal T}_{0}|_{I^{\prime}}|\geq 2$. Without loss of generality, we may assume that

[TABLE]

Then \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{4}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{7}=\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{5}+\mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{6}, a contradiction to Lemma II.1.

• 1.1.b)

If $|{\cal T}_{0}|=2$, we may assume that

[TABLE]

By Lemma II.1 and the fact that $|\{0,1\}^{2}|=4$, we must have $|{\cal T}_{0}|_{I^{\prime}}\cap{\cal Y}_{0}|_{I^{\prime}}|\leq 1$, $|{\cal T}_{0}|_{I^{\prime}}\cap{\cal Y}_{1}|_{I^{\prime}}|=1$ and $|{\cal Y}_{0}|_{I^{\prime}}\cap{\cal Y}_{1}|_{I^{\prime}}|=1$. Without loss of generality, we may assume that

[TABLE]

Then for any choice of {\cal T}_{1}|_{I^{\prime}}=\{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{j}\}, $j\in\{1,2,3,4\}$, we could have a contradiction to Lemma II.1.

• 1.1.c)

If $|{\cal T}_{0}|=1$, we may assume that

[TABLE]

Similarly, we have $|{\cal T}_{1}|_{I^{\prime}}\cap{\cal Y}_{0}|_{I^{\prime}}|\leq 1$, $|{\cal T}_{1}|_{I^{\prime}}\cap{\cal Y}_{1}|_{I^{\prime}}|=1$ and $|{\cal Y}_{0}|_{I^{\prime}}\cap{\cal Y}_{1}|_{I^{\prime}}|=1$. Without loss of generality, we may assume that

[TABLE]

If {\cal T}_{0}|_{I^{\prime}}=\{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{j}\}, $j\in\{1,2,3\}$, then in a similar manner as the case 1.1.b), we could obtain contradictions to Lemma II.1. If {\cal T}_{0}|_{I^{\prime}}=\{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{4}\}, since ${\cal C}$ is a $2$-signature code, we must have

[TABLE]

which implies that

[TABLE]

Since \{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{1},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{2},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{3},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{4}\}=\{0,1\}^{2}, (25) cannot be achieved, a contradiction to the assumption.

• 1.1.d)

If $|{\cal T}_{0}|=0$, a similar discussion with the case 1.1.a) will lead to a contradiction. 2. 1.2)

$|{\cal Y}_{0}|=3$ and $|{\cal Y}_{1}|=2$. We can discuss in the same way with the case 1.1) and then also get contradictions. 2. 2.

$|{\cal C}_{0}|=|{\cal C}_{1}|=4$. Then $|{\cal Y}_{0}|+|{\cal Y}_{1}|=4$ and $|{\cal T}_{0}|+|{\cal T}_{1}|=4$. We only need to consider the case that $|{\cal Y}_{0}|=|{\cal Y}_{1}|=2$ and $|{\cal T}_{0}|=|{\cal T}_{1}|=2$. Without loss of generality, we may assume that

[TABLE]

By Lemma II.1, we have $|{\cal Y}_{1}|_{I^{\prime}}\cap{\cal Y}_{0}|_{I^{\prime}}|\leq 1$. Then we discuss the following two subcases.

1. 2.1)

${\cal Y}_{1}|_{I^{\prime}}\cap{\cal Y}_{0}|_{I^{\prime}}=\emptyset$. Without loss of generality, we may assume that

[TABLE]

By Lemma II.1, we must have $|{\cal T}_{0}|_{I^{\prime}}\cap{\cal T}_{1}|_{I^{\prime}}|\leq 1$ and $|{\cal T}_{i}|_{I^{\prime}}\cap{\cal Y}_{j}|_{I^{\prime}}|\leq 1$ for $i\neq j\in\{0,1\}$. By discussing the possible choices of ${\cal T}_{1}|_{I^{\prime}}$, that is, \{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{1},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{3}\}, \{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{1},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{4}\}, \{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{2},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{3}\} or \{\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{2},\mathchoice{\mbox{\boldmath\displaystyle a}}{\mbox{\boldmath\textstyle a}}{\mbox{\boldmath\scriptstyle a}}{\mbox{\boldmath\scriptscriptstyle a}}_{4}\}, we could get contradictions to Lemma II.1 from ${\cal T}_{0}|_{I^{\prime}}$. 2. 2.2)

$|{\cal Y}_{1}|_{I^{\prime}}\cap{\cal Y}_{0}|_{I^{\prime}}|=1$. Without loss of generality, we may assume that

[TABLE]

A similar discussion with the case 2.1) will lead to contradictions.

Then we have $A(4,2)\leq 7$. ∎

Acknowledgment

Miao is grateful to Prof. Itzhak Tamo of Tel Aviv University and Prof. Zhiying Wen of Tsinghua University for their insightful discussions on Section VI. All the authors express their sincere thanks to the three anonymous reviewers and Prof. Joerg Kliewer, the Associate Editor, for their valuable comments and suggestions which greatly improved this paper.