Combinatorial bounds for list decoding

Informally, an error-correcting code has “nice” listdecodability properties if every Hamming ball of “large” radius has a “small” number of codewords in it. Here, we report linear codes with non-trivial list-decodability: i.e., codes of large rate that are nicely listdecodable, 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 codewords in every Hamming ball of radius H 1(1 R 1= ) n. This answers the main open question from the work of Elias [8]. 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 [13] in this vein. Specifically, for every " > 0, we present a polynomial time constructible asymptotically good family of binary codes of rate ("4) that can be list decoded in polynomial time from up to a fraction (1=2 ") of errors, using lists of size O(" 2). On the negative side, we show that for every Æ and , there exists < Æ, 1 > 0 and an infinite family of linear codes fCigi such that if ni denotes the block length of Ci , then Ci has minimum distance at least Æ ni and contains more than 1 n i codewords in some Hamming ball of radius ni . 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. Keywords— Error-correcting codes, List decoding, Concatenated codes, Reed-Solomon code.