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As separable code (SC, IEEE Trans Inf Theory 57:4843-4851, 2011) andframeproof code (FPC, IEEE Trans Inf Theory 44:1897-1905, 1998) do inmultimedia fingerprinting, strongly separable code (SSC, Des. Codes andCryptogr.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 multimediafingerprinting since SSC has lower tracing complexity than that of SC (the samecomplexity as FPC) and weaker structure than that of FPC. In this paper, wefirst 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 anSSC and an SC. %Consequently a more tighter upper bound $(3q^2/4)$ and lowerbound $(q^{3/2})$ on the number of codewords are obtained; (2) An improvedlower 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 beforeknown bound $\lfloor\sqrt{q}\rfloor^{3}$, which is obtained by means ofdifference matrix and the known result on the subset of $\mathbb{F}^{n}_q$containing no three points on a line.

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