New upper bounds for parent-identifying codes and traceability codes

In the last two decades, parent-identifying codes and traceability codes are introduced to prevent copyrighted digital data from unauthorized use. They have important applications in the scenarios like digital fingerprinting and broadcast encryption schemes. A major open problem in this research area is to determine the upper bounds for the cardinalities of these codes. In this paper we will focus on this theme. Consider a code of length N which is defined over an alphabet of size q. Let M_IPPC(N, q, t) and M_TA(N, q, t) denote the maximal cardinalities of t-parent-identifying codes and t-traceability codes, respectively, where t is known as the strength of the codes. We show M_IPPC(N, q, t) ≤ rq^{⌈N/(v-1)⌉}+ (v- 1 - r) q^{⌊N/(v-1)⌋}, where v= ⌊ (t/ 2 + 1) ²⌋ , 0 ≤ r≤ v- 2 and N≡rmod(v-1). This new bound improves two previously known bounds of Blackburn, and Alon and Stav. On the other hand, M_TA(N, q, t) is still not known for almost all t. In 2010, Blackburn, Etzion and Ng asked whether MTA(N,q,t)≤cq⌈N/t2⌉ or not, where c is a constant depending only on N, and they have shown the only known validity of this bound for t= 2. By using some complicated combinatorial counting arguments, we prove this bound for t= 3. This is the first non-trivial upper bound in the literature for traceability codes with strength three.