Combinatorial bounds for list decoding

Informally, an error-correcting code has "nice" list-decodability properties if every Hamming ball of "large" radius 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 radius H^-1(1-R-1/c)·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 list 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 ε > 0, we present a polynomial time constructible asymptotically good family of binary codes of rate Ω(ε⁴) that can be list-decoded in polynomial time from up to a fraction (1/2-ε) of errors, using lists of size O(ε^-2). On the negative side, we show that for every δ and c, there exists τ < δ, c₁ > 0, and an infinite family of linear codes {C_i}_i such that if n_i denotes the block length of C_i, then C _i has minimum distance at least δ · n_i and contains more than c₁ · n_i^c codewords in some Hamming ball of radius τ · n_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