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

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 $Ω(q^{5/3}+q^{4/3}-q)$ 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.