Limits to list decodability of linear codes

Abstract

We consider the problem of the best possible relation between the list decodability of a binary linear code and its minimum distance. We prove, under a widely-believed number-theoretic conjecture, that the classical gives, in general, the best possible relation between the list decoding radius of a code and its minimum distance. The analogous result is known to hold by a folklore random coding argument for the case of non-linear codes, but the linear case is more subtle and has remained open.We prove our result by exhibiting an infinite family of binary linear codes of minimum distance with a super-polynomial number (in blocklength) of codewords all within a Hamming ball of radius close to the Johnson bound. Even the existence of codes with a super-polynomial number of codewords in a ball of radius bounded away from the minimum distance (let alone radius close to the Johnson bound) was open prior to our work. We also unconditionally prove the tightness of the Johnson bound for decoding with list size that is an arbitrarily large constant.