Combinatorial bounds for list decoding

Abstract

Informally, an error-correcting code has nice list-decodability properties if every Hamming ball of has a small number of codewords in it. We report linear codes with nontrivial list-decodability: i.e., codes of large rate that are nicely list-decodable, and codes of large distance that are not nicely list-decodable. Specifically, on the positive side, we show that there exist codes of rate R and block length n that have at most c codewords in every Hamming ball of H/sup -1/(1-R-1/c)/spl middot/n. This answers the main open question from the work of Elias (1957). This result also has consequences for the construction of concatenated codes of good rate that are decodable from a large fraction of errors, improving previous results of Guruswami and Sudan (see IEEE Trans. Inform. Theory, vol.45, p.1757-67, Sept. 1999, and Proc. 32nd ACM Symp. Theory of Computing (STOC), Portland, OR, p. 181-190, May 2000) in this vein. Specifically, for every /spl epsi/ > 0, we present a polynomial time constructible asymptotically good family of binary codes of rate /spl Omega/(/spl epsi//sup 4/) that can be list-decoded in polynomial time from up to a fraction (1/2-/spl epsi/) of errors, using lists of size O(/spl epsi//sup -2/). On the negative side, we show that for every /spl delta/ and c, there exists /spl tau/ 0, and an infinite family of linear codes {C/sub i/}/sub i/ such that if n/sub i/ denotes the block length of C/sub i/, then C/sub i/ has minimum distance at least /spl delta/ /spl middot/ n/sub i/ and contains more than c/sub 1/ /spl middot/ n/sub i//sup c/ codewords in some Hamming ball of /spl tau/ /spl middot/ n/sub i/. While this result is still far from known bounds on the list-decodability of linear codes, it is the first to bound the radius for list-decodability by a polynomial-sized list away from the minimum distance of the code.