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. 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 ⌈ ℓ / 4 ⌉ codewords. The paper shows that this construction is close to best possible: there exists a constant c, depending only on ℓ, such that a q-ary 2-traceability code of length ℓ contains at most c q ⌈ ℓ / 4 ⌉ codewords. This answers a question of Kabatiansky from 2005. nd 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 ⩽ k 2 . These parameters are of interest since the error correcting construction cannot be used to construct k-traceability codes of constant rate for these parameters: suitable error correcting codes do not exist when q ⩽ k 2 because of the Plotkin bound. Kabatiansky (2004) answered Barg and Kabatianskyʹs question (positively) in the case when k = 2 . This result is generalised to the following: whenever k and q are fixed integers such that k ⩾ 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.