We construct 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 on 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 second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield. In both cases, we develop a linear-algebraic approach to perform list decoding, which pins down the candidate messages to a subspace with a nice “periodic” structure. To prune this subspace and obtain a good bound on the list-size, we pick subcodes of these codes by pre-coding into certain subspace-evasive sets which are guaranteed to have small intersection with the sort of periodic subspaces that arise in our list decoding. We develop two approaches for constructing such subspace-evasive sets. The first is a Monte Carlo construction of hierearchical subspace-evasive (h.s.e) sets which leads to excellent list-size but is not explicit. The second approach exploits a further ultra-periodicity of our subspaces and uses a novel construct called subspace designs, which were subsequently constructed explicitly and also found further applications in pseudorandomness. To get a family of codes over a fixed alphabet size, we instantiate our approach with algebraic-geometric codes based on the Garcia-Stichtenoth tower of function fields. Combining this with pruning via h.s.e sets yields codes list-decodable up to a 1 ? R ? ε error fraction with list size bounded by O(1/ε), matching the existential bound for random codes up to constant factors. Further, the alphabet size can be made exp(O?(1/ε2 )) which is not much worse than the lower bound of exp(?(1/ε)). The parameters we achieve are thus quite close to the existential bounds in all three aspects—error-correction radius, alphabet size, and list-size— simultaneously. This construction is, however, Monte Carlo and the claimed list decoding property only holds with high probability. Once the code is (efficiently) sampled, the encoding/decoding algorithms are deterministic with a running time Oε(N c ) for an absolute constant c, where N is the code’s block length. Using subspace designs instead for the pruning, our approach yields a deterministic construction of an algebraic code family of rate R with efficient list decoding from 1?R?ε fraction of errors over an alphabet of constant size exp(O?(1/ε2 )). The list size bound is upper bounded by a very slowly growing function of the block length N; in particular, it is at most O(log(r) N) (the r’th iterated logarithm) for any fixed integer r. The explicit construction avoids the shortcoming of the Monte Carlo sampling at the expense of a worse list size.
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