Traceability codes, introduced by Chor, Fiat, and Naor in 1994, are combinatorial structures designed for traitor tracing schemes to protect digital content. A t-traceability code enables the identification of the source of digital content, assuming no more than t users have colluded. A major open problem in this research area is to determine the cardinalities of these codes. Let \(M_{TA}(n, q, t )\) denote the maximal cardinality of q-ary t-traceability codes of length n. \(M_{TA}(n, q, t )\) is still not known for almost all t. Blackburn, Etzion and Ng (2010) asked whether \(M_{TA}(n, q, t ) \le c q^{\lceil n/t^2\rceil }\) for some constant c depending only on n and t. The only known validated cases of this bound are \(t = 2\) and 3, which have been proven by Blackburn, Etzion and Ng (2010) and Shangguan, Ma and Ge (2018), respectively. In this paper, we establish key lemmas applicable to traceability codes of any strength level and use them to derive an upper bound that addresses the question when the minimum distance is sufficiently large. In particular, these lemmas allow us to establish upper bounds for t-traceability codes when \(t=3\) and \(t=4\) that positively answer the question.
