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 \(M_{IPPC}(N,q,t)\le rq^{\lceil N/(v-1)\rceil }+(v-1-r)q^{\lfloor N/(v-1)\rfloor }\), where \(v=\lfloor (t/2+1)^2\rfloor \), \(0\le r\le v-2\) and \(N\equiv r \mod (v-1)\). 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 \(M_{TA}(N,q,t)\le cq^{\lceil N/t^2\rceil }\) 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.