Optimal rate list decoding over bounded alphabets using algebraic-geometric codes
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We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction 1-R-ϵ of adversarial errors where R is the rate of the code, for any desired positive constant ϵ. The alphabet size depends only ϵ and is nearly-optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The list decoding algorithm is based on a linear-algebraic approach, which pins down the candidate messages to a subspace with a nice "periodic" structure. The list is pruned by precoding into a special form of "subspace-evasive" sets, which are constructed pseudorandomly. Instantiating this construction with the Garcia-Stichtenoth function field tower yields codes list-decodable up to a 1-R-ϵ error fraction with list size bounded by O(1/ϵ), matching the existential bound up to constant factors. The parameters we achieve are thus quite close to the existential bounds in all three aspects: error-correction radius, alphabet size, and list-size. The second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield. Once again, the linear-algebraic approach to list decoding to pin down candidate messages to a periodic subspace. We develop an alternate approach based on "subspace designs" to precode messages. Together with the subsequent explicit constructions of subspace designs, this yields a deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-ϵ fraction of errors over a constant-sized alphabet. The list size is bounded by a very slowly growing function of the block length N; in particular, it is at most O(^(r) N) (the r'th iterated logarithm) for any fixed integer r.
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