Bounds and Constructions for $\overline{3}$-Strongly Separable Codes with Length $3$

TL;DR

This paper establishes bounds and constructions for $ar{3}$-strongly separable codes of length 3, improving understanding of their size and structure for multimedia fingerprinting applications.

Contribution

It derives upper bounds on the size of $ar{t}$-SSC and provides a tighter bound and equivalence results specifically for $ar{3}$-SSC with length 3, including new lower bounds.

Findings

Derived tighter upper bounds on codeword numbers for $ar{t}$-SSC.

Established an equivalence between SSC and SC for $ar{3}$-SSC.

Provided improved lower bounds for the size of $q$-ary $ar{3}$-SSC when $q=q_1^6$.

Abstract

As separable code (SC, IEEE Trans Inf Theory 57:4843-4851, 2011) and frameproof code (FPC, IEEE Trans Inf Theory 44:1897-1905, 1998) do in multimedia fingerprinting, strongly separable code (SSC, Des. Codes and Cryptogr.79:303-318, 2016) can be also used to construct anti-collusion codes. Furthermore, SSC is better than FPC and SC in the applications for multimedia fingerprinting since SSC has lower tracing complexity than that of SC (the same complexity as FPC) and weaker structure than that of FPC. In this paper, we first derive several upper bounds on the number of codewords of $\overline{t}$-SSC. Then we focus on $\overline{3}$-SSC with codeword length $3$, and obtain the following two main results: (1) An equivalence between an SSC and an SC. %Consequently a more tighter upper bound $(3q^2/4)$ and lower bound $(q^{3/2})$ on the number of codewords are obtained; (2) An improved lower bound $\Omega (q^{5/3}+q^{4/3}-q)$ on the size of a $q$-ary SSC when $q=q_1^6$ for any prime power $q_1\equiv\ 1 \pmod 6$, better than the before known bound $\lfloor\sqrt{q}\rfloor^{3}$, which is obtained by means of difference matrix and the known result on the subset of $\mathbb{F}^{n}_q$ containing no three points on a line.