Good traceability codes do exist

Traceability codes are combinatorial objects introduced by Chor, Fiat and Naor in 1994 to be used to trace the origin of digital content in traitor tracing schemes. Let $F$ be an alphabet set of size $q$ and $n$ be a positive integer. A $t$-traceability code is a code $\mathscr{C}\subseteq F^n$ which can be used to catch at least one colluder from a collusion of at most $t$ traitors. It has been shown that $t$-traceability codes do not exist for $q\le t$. When $q>t^2$, $t$-traceability codes with positive code rate can be constructed from error correcting codes with large minimum distance. Therefore, Barg and Kabatiansky asked in 2004 that whether there exist $t$-traceability codes with positive code rate for $t+1\le q\le t^2$. In 2010, Blackburn, Etzion and Ng gave an affirmative answer to this question for $q\ge t^2-\lceil t/2\rceil+1$, using the probabilistic methods. However, they did not see how their probabilistic methods can be used to answer this question for the remaining values of $q$. They even suspected that there may be a `Plotkin bound' of traceability codes that forbids the existence of such codes. In this paper, we give a complete answer to Barg-Kabatiansky's question (in the affirmative). Surprisingly, our construction is deterministic.