Traceability codes

Traceability codes are combinatorial objects introduced by Chor, Fiat and Naor in 1994 to be used in traitor tracing schemes to protect digital content. A k-traceability code is used in a scheme to trace the origin of digital content under the assumption that no more than k users collude. It is well known that an error correcting code of high minimum distance is a traceability code. When does this 'error correcting construction' produce good traceability codes? The paper explores this question. Let @? be a fixed positive integer. When q is a sufficiently large prime power, a suitable Reed-Solomon code may be used to construct a 2-traceability code containing q^@?^@?^/^4^@? codewords. The paper shows that this construction is close to best possible: there exists a constant c, depending only on @?, such that a q-ary 2-traceability code of length @? contains at most cq^@?^@?^/^4^@? codewords. This answers a question of Kabatiansky from 2005. Barg and Kabatiansky (2004) asked whether there exist families of k-traceability codes of rate bounded away from zero when q and k are constants such that q==2 and q>=k^2-@?k/2@?+1, or such that k=2 and q=3, there exist infinite families of q-ary k-traceability codes of constant rate.