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 ℓ 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 cq⌈ℓ/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⩽k2. 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⩽k2 because of the Plotkin bound. Kabatiansky (2004) answered Barg and Kabatiansky's 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⩾k2−⌈k/2⌉+1, or such that k=2 and q=3, there exist infinite families of q-ary k-traceability codes of constant rate.