Signature codes for weighted binary adder channel and multimedia fingerprinting
Jinping Fan, Yujie Gu, Masahiro Hachimori, and Ying Miao

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 denote the maximum cardinality of a -signature code of length , and denote the maximum cardinality of a -signature code of length and constant weight . First, we derive asymptotic and general upper bounds of by relating signature codes to 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 and for infinitely many by connecting signature codes with -free graphs and union-free families, respectively. Third, we provide two explicit constructions for -signature codes which have efficient decoding algorithms and applications to two-level signature codes.…
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Signature codes for weighted binary adder channel and multimedia fingerprinting
††thanks: J. Fan is with the Department of Policy and Planning Sciences, Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan (e-mail: [email protected]). ††thanks: Y. Gu is with the Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv, Israel and the Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan (e-mail: [email protected]). ††thanks: M. Hachimori is with the Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan (e-mail: [email protected]). ††thanks: Y. Miao is with the Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan (e-mail: [email protected]). Research supported by JSPS Grant-in-Aid for Scientific Research (B) under Grant No. 18H01133.
Jinping Fan, Yujie Gu, Masahiro Hachimori, and Ying Miao
Abstract
In this paper, we study binary signature codes for the weighted binary adder channel (WbAC) and collusion-resistant multimedia fingerprinting. Let denote the maximum size of a -signature code of length , and denote the maximum size of a -signature code of length and constant-weight . First, we derive asymptotic and general upper bounds on by relating signature codes to 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 and for infinitely many by connecting signature codes with -free graphs and union-free families, respectively. Third, we provide two explicit constructions for -signature codes which have efficient decoding algorithms and applications to two-level signature codes. Furthermore, we show from a 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.
Index Terms:
Signature code, weighted binary adder channel, multimedia fingerprinting
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, , called the host signal. To prevent unauthorized redistribution of outside of authorized users, the dealer constructs a set of watermarks, also called fingerprints, using a linear modulation scheme based on 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 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 -th authorized user, , is represented as
[TABLE]
where for antipodal modulation and for on-off keying type of modulation [39]. In this paper, we concentrate on the on-off keying type of modulation, that is, . 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 , \{\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 , \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 -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 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}, . In a linear attack, a coalition 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 , that is,
[TABLE]
where , and for all 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 are chosen by the coalition but unknown to the dealer. When and for all , 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 where 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 for any . To resist the linear attack, the dealer aims to identify the coalition based on the result even without any knowledge of , , 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 was considered, which can be represented by a mapping on : \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 . Since and for all , by (3), it is easy to see that for any . 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 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 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 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 based on the result even though , 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 .
In [23], Egorova et al. showed that this problem is in fact equivalent to the problem of designing a -signature code for the weighted binary adder channel (WbAC). Suppose that users would like to communicate with the same destination through a shared WbAC in the multiple-access communication system, among which at most users are active simultaneously. To communicate successfully, each user 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}, . If the users in are active at the same time, they input their vectors simultaneously into the WbAC. The output in the destination is a vector as in (4), where 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 using the corresponding channel output , 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 , are real numbers in WbAC was first investigated by Mathys [41]. Egorova et al. [23] first considered the scenario that , are real numbers such that . 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 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 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 -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 codes are also presented. In Section III, we first relate -signature codes to bipartite graphs without cycles of length less than or equal to and then obtain general upper bounds for -signature codes of length from the known results on bipartite graphs. Asymptotic upper bounds for -signature codes are obtained based on the known results on codes. In Section IV, we first show that a -signature code with constant-weight is equivalent to a -free graph, yielding exact values of and optimal constructions for -signature codes with constant-weight for infinite values of . Exact values of are also derived and optimal constructions for -signature codes with constant-weight are obtained for all with a few exceptions by investigating their combinatorial properties. In Section V, we provide two explicit constructions for -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 codes are also presented.
II-A Signature code
Let and be positive integers, and 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 -ary code of length and size , or 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 is called a codeword and is called the -th coordinate of \mathchoice{\mbox{\boldmath\displaystyle c}}{\mbox{\boldmath\textstyle c}}{\mbox{\boldmath\scriptstyle c}}{\mbox{\boldmath\scriptscriptstyle c}}_{j}. Let be an code. Define as the code rate of .
The matrix representation of is an matrix where each column vector is a codeword of . Denote . For any 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 restricted to by deleting all the coordinates of with , 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 restricted to .
An code is also called a binary code. Let 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 . Denote as the family of all subsets of , and as the family of all -subsets of . has constant-weight 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 be an code and be an integer. is an -signature code if*
for any two distinct subsets with , we have
[TABLE]
for any 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.
We first generalize the definition of signature code by considering multi-subsets of 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 , we call the set formed by all the elements in as the base set of . 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 of a set is a multi-set such that the base set of is a subset of .
Example 1
Let be a set. Then and are two distinct multi-subsets of with the same base set . Let . Then the base set of is which is distinct to that of and .
We have the following equivalent but more general description of Definition II.1.
Proposition II.1
An code is a -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 with and distinct base sets, we have
[TABLE]
for any real numbers such that .
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 -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 with and distinct base sets, and also exist with such that
[TABLE]
By combining like terms in the left-hand and right-hand sides of (5) respectively, we can obtain
[TABLE]
where are base sets of respectively satisfying and , and 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 is a -signature code (see Definition II.1). The conclusion follows.
∎
Proposition II.1 shows that in Definition II.1, the condition “two distinct subsets with ” could be equivalently replaced by “two multi-subsets of size no more than with distinct bases sets”. We remark that in the remainder of this paper, always refer to subsets unless otherwise specified.
II-B A geometric perspective
Signature codes can be interpreted geometrically. Let be the vertex-set of an -dimensional hypercube. Then a binary code of length corresponds to a subset of vertices of an -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 as
[TABLE]
For any distinct , the open convex polytopes of and do not cross if
[TABLE]
Then we have
Proposition II.2
An code is a -signature code if and only if for any distinct subsets with , we have .
Proposition II.2 can be immediately derived from Definition II.1. The following example illustrates our argument.
Example 2
Consider the vertices of the -dimensional hypercube as shown in Figure 1. The open convex polytope formed by any two distinct vertices is the line segment between and except themselves. Let . It is easy to check that the open convex polytopes formed by any two vertices in do not cross.
Then the set of vectors corresponding to , that is, , is a -signature code. The matrix representation of is presented below.
[TABLE]
In the application of an -signature code to multimedia fingerprinting, suppose that corresponds to the set of fingerprints distributed to a coalition set . As introduced in Section I, once a forged copy produced by the coalition is confiscated, the dealer can calculate a result . By (3), (4) and (6), is a point in . Then we can trace back to , or equivalently the coalition set , by checking that to which open convex polytope, formed by a subset of with size no more than , belongs. The computational complexity of the tracing algorithm is .
II-C Known results
Let denote the maximum size of a -signature code of length , and denote the maximum size of a -signature code of length and constant-weight . A -signature code of length is called optimal if it has size , and a -signature code of length and constant-weight is called optimal if it has size . Denote the largest asymptotic code rate of -signature codes as
[TABLE]
In [23], Egorova et al. found that a binary code with any codewords linearly independent is a -signature code. Based on this observation, they constructed signature codes from binary Goppa codes in coding theory and obtained a lower bound on .
Theorem II.1
([23])**
[TABLE]
This immediately implies
Corollary II.2
Let be an integer. Then
In the literature, the exact values of and are generally unknown except the lower bounds in Theorem II.1 and Corollary II.2. However, the lower bound on in Theorem II.1 is not tight in general since for instance it shows , while Example 2 implies . The main purpose of this paper is to explore bounds on and by using combinatorial methods, and to provide constructions for -signature codes which are equipped with efficient tracing algorithms.
II-D Relationships with codes
In this subsection, we build relationships between signature codes and codes.
The investigation of code was motivated by -sequence introduced by Erdős and Turán [26]. codes for were first studied by Lindström in [37] and later in [38] under the name of -sequences of vectors. In recent decades, 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 code is a code, or 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 , we have
[TABLE]
From Definition II.2, it is obvious that when and , an code is 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 codes.
Lemma II.1
Let be an code and be an integer. If is a -signature code, then is a 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 code. Assume that is a -signature code, but not a 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 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 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 which satisfy , \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 are distinct. Let for any . Then we have . 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 is a -signature code by Proposition II.1. The lemma follows. ∎
The converse of Lemma II.1 does not hold in general. However, when , we have the following observation.
Lemma II.2
Let be an code. Then is a -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, is a -signature code if and only if it is a 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 is not a -signature code. Then according to Definition II.1, there exist two distinct subsets with , and also exist 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 or , then we must have , a contradiction to the assumption. 2. 2.
If and , then we can have , a contradiction. 3. 3.
If and , 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 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 . For any , we discuss the following three subcases of .
- 3.a)
If , then , which implies since . 2. 3.b)
If , then since , which implies since and . 3. 3.c)
If , then , which implies .
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 -signature codes
In this section, we investigate upper bounds on and . First, we derive a general upper bound on by connecting a -signature code with a bipartite graph containing no cycles of length less than or equal to , which also implies an upper bound on . Second, we give an improved upper bound for based on the known results on codes.
III-A General upper bounds on -signature codes
Denote as a bipartite graph where are two disjoint sets of vertices and is the set of edges each connecting one vertex in and the other in . is called complete if every vertex in is connected to every vertex in . A complete bipartite graph with and is denoted as . A cycle of is a sequence of vertices , where , are all distinct and for any . The number of vertices in a cycle is called the length of this cycle, and the girth of a graph is the length of the shortest cycle in . If a graph contains no cycle of length , it is called -free. A matching of is a subset of edges without common vertices, and a perfect matching is a matching which matches all vertices in .
Let be an code. For a partition of with , and , we define a bipartite graph corresponding to as such that , 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 as the number of edges in . It is obvious that , and . Moreover, we have the following observation.
Lemma III.1
If is a -signature code, then for any partition of , the girth of the corresponding bipartite graph is at least .
Proof:
Note that a bipartite graph contains no cycles of odd length. To show has girth at least , it suffices to show that contains no cycles of length for any . Assume that is a -signature code, but contains a cycle of length for some . Without loss of generality, we may assume that the cycle is , where and . Let for , and . Then and are two disjoint perfect matchings of the cycle . Recall that each edge of corresponds to a codeword of . 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 in for , 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 and are two disjoint subsets of such that , \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 is a -signature code. The conclusion follows. ∎
The following result on bipartite graphs was shown in [42].
Lemma III.2
([42])* Let be a bipartite graph with and . If is -free, then*
[TABLE]
By Lemmas III.1 and III.2, we obtain
Theorem III.1
Let be integers. Then
[TABLE]
Proof:
Let be a -signature code of length and size . By Lemma III.1, for any partition of with , and , the corresponding bipartite graph with , and is -free for any . Then by Lemma III.2, we have that
[TABLE]
holds for any integers such that and . To make the upper bound on 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 , Theorem III.1 shows that . A better upper bound for could be obtained by using Roman’s bound [44] on bipartite graphs.
Lemma III.3
([44])* Let be a bipartite graph with , , and let be an integer such that . If G contains no , then*
[TABLE]
By Lemmas III.1 and III.3, we have
Theorem III.2
Let be an integer. Then
[TABLE]
Proof:
Let be a -signature code of length and size . Similarly, according to Lemma III.1, for any partition of with , and , the corresponding bipartite graph with , and is -free. Note that . Then by choosing and in Lemma III.3, we have that
[TABLE]
holds for any integers such that and . To make the upper bound on as small as possible, we take
[TABLE]
Then (8) follows.
∎
It can be checked that the upper bound on in Theorem III.2 is tighter than that in Theorem III.1 for the case . Indeed, for instance Theorem III.1 tells while Theorem III.2 shows . The following result shows that the upper bound on in Theorem III.2 is tight for certain cases.
Corollary III.3
*, and . *
Proof:
For , it is easy to check that is a -signature code. Moreover, Theorem III.2 shows that , resulting .
For , Theorem III.2 shows that which, together with Example 2, yields .
For , we have that is a -signature code, implying . Notice that Theorem III.2 only tells that . However, we can show whose proof is deferred to Appendix, and thus we have . ∎
From Theorem III.1, we can obtain an upper bound on for any integer . Here, we remark that although the upper bound on in Theorem III.2 is tighter than that in Theorem III.1 for , they imply the same upper bound on the largest asymptotic code rate . In general, by Theorem III.1, we have
Corollary III.4
Let be an integer. Then
[TABLE]
III-B Asymptotic upper bounds on -signature codes
In this subsection, we provide an improved upper bound on based on Lemma II.1 and the known results on codes.
Denote as the maximum size of a binary code of length . Define the largest asymptotic code rate of binary 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 . The best known upper bounds on are as follows.
Lemma III.4
Let be an integer. Then
([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 , .*
It is obvious from Lemmas II.1 and II.2 that for any integer and the equality holds for .
Combining this with Lemma III.4, we obtain
Theorem III.5
Let be an integer. Then
* and .* 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 , .
We remark that the upper bound of obtained in Theorem III.5 is tighter than that obtained in Corollary III.4. For example, Corollary III.4 implies and , while Theorem III.5 shows that and .
IV Optimal -signature codes with constant-weights and
In this section, we study the combinatorial properties of -signature codes of length and constant-weights and , respectively. Accordingly, bounds on and are obtained, and optimal -signature codes of length and constant-weights and for infinite values of are constructed respectively.
IV-A Optimal -signature codes with constant-weight
Let be a binary code of length and constant-weight . Let be the corresponding graph of where and for any distinct , 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
* is a -signature code of constant-weight if and only if its corresponding graph is -free.*
Proof:
We first show the necessity. If there exist four vertices which form a cycle in graph , 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 of is -free, but is not a -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 in , a contradiction to the assumption. The lemma follows. ∎
Lemma IV.1 shows that if there exists a -signature code with constant-weight , then we can construct a -free graph, and conversely, given a -free graph, we can obtain a -signature code with constant-weight . The following example illustrates this relationship.
Example 3
Consider the -free graph on vertices depicted in Figure 2.
Then according to Lemma IV.1, we can obtain a -signature code of length and constant-weight as follows.
[TABLE]
Let denote the maximum number of edges in a -free graph with vertices. Then by Lemma IV.1, we have . It was proved in [6, 25] that is asymptotically equal to . In [28, 29, 30], it was shown that if and is a power of or a prime power greater than , then and a -free graph with vertices and edges could be constructed from the polarity of a projective plane of order , see [25, 28, 29, 30] for more details. For any integer , the exact value of and the corresponding -free graph with vertices and edges were presented in [13]. Based on these known results for -free graphs and Lemma IV.1, we can obtain the following results for -signature codes with constant-weight .
Theorem IV.1
Let be an integer.
* is asymptotically equal to .* 2. 2.
If where is a power of or is a prime power, then and an optimal -signature code of length and constant-weight can be constructed from the polarity of a projective plane of order . 3. 3.
For , the exact values of are listed in Table I, and optimal -signature codes of length and constant-weight can be constructed from the corresponding -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 -free graphs outlined above. Besides, the -free graph depicted in Figure 2 was presented in [13]. By Theorem IV.1, the obtained -signature code with constant-weight in Example 3 is optimal.
IV-B Optimal -signature codes with constant-weight
In this subsection, we investigate the combinatorial properties of -signature codes of constant-weight by means of Frankl and Füredi’s method in [27], where they studied binary codes using techniques in extremal set theory.
Let be a binary code of length and constant-weight . Notice that can be uniquely described as a family 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 , denote
[TABLE]
Then we have the following observation.
Lemma IV.2
* is a -signature code of constant-weight if and only if for any distinct .*
Proof:
To show the necessity, assume that there exist distinct and also distinct elements such that . Then are four distinct members of . Suppose that their corresponding codewords in 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 satisfies that for any distinct , but is not a -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}), which are four distinct members of . Suppose that and . If , then we must have 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 . Now we discuss the following two cases.
If , assume that . Then . 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 . Thus one of and should be in and the other in , and so as and . Without loss of generality, we may assume that and . Let and . Then , a contradiction to that . 2. 2.
If , then . Without loss of generality, we may assume that and . Let and . Then , also a contradiction.
The conclusion then follows. ∎
Based on Lemma IV.2, we shall derive an upper bound for 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
* for any .*
Proof:
Let be an -signature code of constant-weight with . According to (9), we have
[TABLE]
since each -elements set has subsets of size and each codeword in is calculated three times in the right-hand side of (10). Besides, Lemma IV.2 tells that for any distinct , implying
[TABLE]
Note that the maximum value of under the condition (11) is achieved when for all . Hence by (10),
[TABLE]
and the theorem follows. ∎
Next, we show that the upper bound on in Theorem IV.2 can be achieved for all values of with some exceptions by establishing the relationship between signature codes and weakly union-free families. A family is called weakly union-free if for any distinct , .
Lemma IV.3
Let be a binary code of length and be the corresponding family. If is weakly union-free, then is a -signature code.
Proof:
Assume that is weakly union-free, but is not a -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 is a -signature code, the corresponding family may not be weakly union-free. The following is an example.
Example 4
Let be a binary code of length and constant-weight defined as below.
[TABLE]
Its corresponding family is . It is easy to check that is a -signature code, but is not weakly union-free since .
Denote as the maximum size of a weakly union-free family and a weakly union-free family is called optimal if it has size . Then by Lemma IV.3, we have . In [15, 8, 27], the exact value of for any and was proved and direct constructions from combinatorial design theory for optimal weakly union-free families were presented. Combining this with Theorem IV.2 and Lemma IV.3, we have
Theorem IV.3
* for any and .*
We remark that by Theorem IV.3, optimal -signature codes of length and constant-weight for and can be constructed from optimal weakly union-free families . The interested reader is referred to [15, 8, 27] for more details on the constructions for optimal weakly union-free families .
V Constructions for -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 be an code. For any , 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 -th coordinates of . The descendant code of is defined as
[TABLE]
Definition V.1
Let be an integer.
* is an -frameproof code, or an -FPC, if for any subset with , we have , 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 such that .* 2. 2.
* is an -superimposed code, if for any subset with and any other codeword , there exists such that and .*
V-A Concatenated construction
We now give the concatenated construction by taking a small -signature code as the inner code which guarantees the concatenated code to be a -signature code, and taking a -ary -FPC as the outer code which makes sure the concatenated code can reduce complexity of decoding compared to an arbitrary -signature code of the same length and size.
Construction 1
Let be an -FPC over the alphabet and be an -signature code. Define a bijection . Let be the code defined by
[TABLE]
Then is an -signature code.
Proof:
It is obvious that is an code. For any distinct with , and for any 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, we would like to show
[TABLE]
Since , without loss of generality, we may assume that there exists one codeword 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 correspond to respectively, and 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, , 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 is an -FPC, there must exist one coordinate , such that . This implies that . Let . 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 with distinct base sets. Since the inner code is a -signature code, by Proposition II.1 and the fact that , we have
[TABLE]
for any 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, which implies (12). Hence is a -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 -FPC constructed from an orthogonal array in [3] as the outer code and use a -signature code of length shown in Corollary III.3 as the inner code.
Example 5
Let be a -FPC and be a -signature code defined below.
[TABLE]
Define the bijection from 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 based on the outer code and inner code is a -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 in Example 5 is , while the code rate of the inner code is . 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 obtained by Construction 1 can trace back to a coalition of size at most in time .
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 . Suppose that , corresponds to the real coalition for the output . To determine by and the code , 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}, , a subset , can be traced back since the inner code is a -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, . The time cost in this step is .
Step 2. Next we decode the outer code. For the outer code , 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 for all . 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 .
Now we show that will be identified after Steps 1 and 2. To this end, we show that . By Step 1, we have . 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 is a -FPC. Hence we have .
Based on Steps 1 and 2, the concatenated signature code can trace back to a coalition of size at most in time . ∎
In general, as described in Section II, the decoding complexity of an -signature code is . According to Theorem V.1, the -signature code obtained by Construction 1 can reduce the decoding complexity to if we choose an -FPC with as the outer code. This can be achieved due to the known results on -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 -signature code in Example 5 being applied to construct fingerprints for 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 , . 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 is the output of multimedia fingerprinting channel generated by 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 by and in the following two steps.
Step 1. As stated in Example 5, the inner code in (13) is a -signature code. For each \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}, , we can decode that
[TABLE]
where , and .
Step 2. Denote as the subset of the outer code corresponding to . Then by (13) and (14), we obtain
[TABLE]
To determine , it is equivalent to determine . As stated in Example 5, the outer code is a -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}, , such that for any .
V-B Product construction
Now we provide a product construction by combining superimposed codes and signature codes as follows. The ingredient -signature code makes the constructed code -signature, and the ingredient -superimposed code allows for efficient decoding. Our constructed code can reduce decoding complexity compared to an arbitrary -signature code of the same length and size.
Construction 2
Let be an -superimposed code, and be an -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 , where is the Kronecker product of and :
[TABLE]
Then is an -signature code.
Proof:
It is obvious that is an code, and thus is an code. For simplicity, we divide into groups of codewords as
[TABLE]
where
[TABLE]
For any two distinct subsets such that and , denote and . Obviously, and . We would like to show that for any 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, we have
[TABLE]
To this end, we consider the following two cases.
and , .
- 1.a)
If , without loss of generality, we may assume . Then . Since is an -superimposed code, does not have as a column vector. Then there must exist , such that . Denote , . Recall that . 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 . Since is a -signature code, we have
[TABLE]
for any 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, implying (16). 2. 1.b)
If , without loss of generality, we may assume . Then and for any . Since , there exists at least one such that . Without loss of generality, we may assume . Since is a -superimposed code, there exists , such that and for any . 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}, . 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 -signature code, we have
[TABLE]
for any 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}\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 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.
. Without loss of generality, we may assume and . Since is a -superimposed code, there exists , such that and for all . By the construction of 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}}\}, contains at least one nonzero vector and , 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 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 is a -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 identity matrix, which is a -superimposed code by Definition V.1, and the -signature code shown in Example 2 are employed.
Example 7
Let be a -superimposed code and be a -signature code defined below.
[TABLE]
Then by Construction 2, is a -signature code presented below.
[TABLE]
We remark that the signature code obtained from Construction 2 has larger length and code size than the ingredient signature code , but may be not good in the sense of code rate. For example, the code rate of the signature code in Example 7 is , while the code rate of the ingredient signature code is . 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 obtained by Construction 2 can trace back to a coalition of size at most in time .
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 . Suppose that , corresponds to the real coalition for the output . Denote as the set of group indices of the codewords in . Obviously, . To determine by and the code , we will first determine , and then determine for any .
Step 1. For the ingredient superimposed code , detect each codeword , by checking if there exists , 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 . 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 . We claim that . The time cost in this step is .
To verify our claim, we first show . Assume that there exists but , that is, . Then there exists , 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 . 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 , that is, , a contradiction to the assumption. So we have . On the other hand, we show . Assume that there exists but . Since is a -superimposed code, there exists , such that and for any . Recall that is the set of group indices of the codewords in . 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 , a contradiction to the assumption that since . So, we have and thus can be determined after Step 1.
Step 2. In this step, we show how to determine for any .
If , without loss of generality, we may assume . Then . Since is a -superimposed code, there exists , such that . 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 is a -signature code, we can determine by \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}, and thus can determine . The time cost in this case is . 2. 2.
If , then we have , and for any . Since is a -superimposed code, for any , there exists , such that and for any . 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 -signature code, we can determine by \mathchoice{\mbox{\boldmath\displaystyle r}}{\mbox{\boldmath\textstyle r}}{\mbox{\boldmath\scriptstyle r}}{\mbox{\boldmath\scriptscriptstyle r}}_{i}, and thus can determine for any . The time cost in this case is .
Based on Steps 1 and 2, the -signature code obtained by Construction 2 can trace back to all the colluders of size no more than in time . ∎
In general, the decoding complexity of an -signature code is . Here according to Theorem V.2, we can reduce the decoding complexity to by using the -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 -signature code in Example 7 being applied to construct fingerprints for 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 , . We divide all the codewords of into groups 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 into , and . Suppose that with represents the coalition set, is the output of multimedia fingerprinting channel generated by , and is the set of group indices of the codewords in . We discuss two cases on the distributions of codewords in .
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 generated by 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 by and in the following two steps.
Step 1. As shown in (17) in Example 7, is a -superimposed code and . 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 , implying that .
Step 2. Based on Step 1, we focus on . As shown in (17), is a -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 generated by 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 by and 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 , we first can decode , implying that .
Step 2. In this step, we determine and . 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 -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 , 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 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 -traceability codes with 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 , which is the same to a traditional -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 but no more than . 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 be an code and be positive integers with . Suppose that all the codewords of are partitioned into groups: . For any subset , denote as the set of group indices of the codewords in . is an two-level -signature code if for any two subsets and for any 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,
[TABLE]
implies that
* if ;* 2. 2.
* if .*
By Definition V.2, in multimedia fingerprinting, a two-level -signature code could
- a)
trace back to all the colluders if the coalition size is at most ; 2. b)
determine all the groups each of which contains at least one colluder if the coalition size is at most .
We have the following relationship between signature codes (one-level) and two-level signature codes.
Lemma V.1
Let be two positive integers such that . Then a -signature code is a two-level -signature code, and a two-level -signature code is a -signature code.
Proof:
By Definitions II.1 and V.2, it is obvious that a two-level -signature code is a -signature code.
Suppose that is a -signature code, then we can partition all the codewords of into non-empty groups.
By Definition II.1 and the condition that , is a -signature code; 2. 2.
For any two subsets , if and the group indices of are different, then . By Definition II.1, for any 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, 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, is a -signature code. The lemma follows. ∎
Lemma V.1 implies that a -signature code lies between a -signature code and a -signature code. The following construction for two-level signature codes is an application of Construction 2.
Theorem V.3
Let be two positive integers such that . Let be an -superimposed code and be an -signature code containing . 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 defined by (15) is an two-level -signature code with groups.
Proof:
By Construction 2, an code can be obtained from and . Now we show that is a two-level -signature code. It is obvious that all the codewords of are divided into groups and each group contains codewords. If a forged copy is created by a coalition of size at most , all the colluders will be identified by Theorem V.2. If a forged copy is created by a coalition of size at most , 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 add some noise 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 . Then the dealer observes the forged copy
[TABLE]
Notice that is chosen by the coalition and unknown to the dealer. However, the dealer can calculate
[TABLE]
for , 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 . It should be noted that the dealer does not know what 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 with some properties to identify the whole coalition set 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 be an code, be an integer and be a real number. has -complete traceability if for any two distinct subsets with , 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 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 -dimensional Euclidean space and be a real number. An open -ball with center and radius 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 , define the distance between and as
[TABLE]
Note that for any distinct , if , then we must have . We have the following equivalent description for Definition VI.1.
Proposition VI.1
An code has -complete traceability if and only if for any two distinct subsets with , we have
[TABLE]
Proof:
Let be an code, be two distinct subsets with and 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 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 . 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 -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 , 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 -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 . Then the argument
[TABLE]
holds if and only if
[TABLE]
Thus, by Definition VI.1, has -complete traceability if and only if for any two distinct subsets with , (23) holds for any 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, that is, according to (6). The conclusion follows.
∎
By Proposition VI.1, in noisy multimedia fingerprinting, a coalition with size no more than can be completely traced back if and only if there exists a binary code such that for any distinct with , the condition (22) is satisfied. However, if , we always have , and thus (22) does not hold. For example, in Example 2, let and . Then and , but 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 be an code, be an integer and be a real number. is an -frameproof signature code if for any two subsets with and , 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 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 be an code, be an integer and be a real number. If has -complete traceability, then is a -frameproof signature code, but not vice versa.
We provide an equivalent description of frameproof signature codes from a geometric viewpoint.
Proposition VI.4
* is an -frameproof signature code if and only if for any with and , we have .*
Example 9
It is easy to see from Example 2 that is a -frameproof signature code.
We remark that although a -frameproof signature code cannot identify any coalition set in noisy multimedia fingerprinting, it can guarantee at least two things:
If is a coalition of size at most , then cannot frame any with and 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 is a coalition of size at most 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 with 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 -signature codes. We explored the combinatorial properties and derived bounds for -signature codes of constant-weights and , respectively. Moreover, we provided two explicit constructions for -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 is a -signature code with . Let
[TABLE]
By Lemma II.1, it is easy to check that (24) is still a -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 -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 for all and .
First we divide into groups with respect to the first and second rows of (24). Let
[TABLE]
If , then is a -signature code of length and , which contradicts Corollary III.3 that . Thus, . Similarly, we also have . Since and , we only need to discuss the following two cases.
and . Then and . By Corollary III.3 that , we have and . There are two subcases.
- 1.1)
and . Without loss of generality, we may assume that
[TABLE]
- 1.1.a)
If , then . Since and , we have . 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 , we may assume that
[TABLE]
By Lemma II.1 and the fact that , we must have , and . 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}\}, , we could have a contradiction to Lemma II.1.
- 1.1.c)
If , we may assume that
[TABLE]
Similarly, we have , and . 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}\}, , 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 is a -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 , a similar discussion with the case 1.1.a) will lead to a contradiction. 2. 1.2)
and . We can discuss in the same way with the case 1.1) and then also get contradictions. 2. 2.
. Then and . We only need to consider the case that and . Without loss of generality, we may assume that
[TABLE]
By Lemma II.1, we have . Then we discuss the following two subcases.
- 2.1)
. Without loss of generality, we may assume that
[TABLE]
By Lemma II.1, we must have and for . By discussing the possible choices of , 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 . 2. 2.2)
. Without loss of generality, we may assume that
[TABLE]
A similar discussion with the case 2.1) will lead to contradictions.
Then we have . ∎
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.
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