Optimal Rate List Decoding over Bounded Alphabets Using Algebraic-geometric Codes.

We give new constructions of two classes of algebraic code families that 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 second class of codes are obtained by restricting evaluation points of an algebraicgeometric 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 that 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 subspaceevasive sets. The first is a Monte Carlo construction of hierearchical subspace-evasive (h.s.e.) sets that 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(Ω(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 (Nc) for an absolute constant c, where N is the code's block length. Using subspace designs instead for the pruning, our approach yields the first 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 slightly worse list size. [ABSTRACT FROM AUTHOR]