Optimal rate list decoding over bounded alphabets using algebraic-geometric codes

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 pre-coding into a special form of subspaceevasive sets. Instantiating this construction with the explicit Garcia-Stichtenoth tower of function fields 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 of the codes is a constant depending only on ε — it can be made exp(Õ(1/ε)) 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. Our code construction is Monte Carlo and has the claimed list decoding property with high probability. Once the code is (efficiently) sampled, the encoding/decoding algorithms are deterministic with a running time Oε(N ) for an absolute constant c, where N is the code’s block length. The second class of codes are obtained by restricting evaluation points of an algebraicgeometric 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 is used to pre-code messages and prune the subspace of candidate solutions. Together with the subsequent explicit constructions of subspace designs, this 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(Õ(1/ε)). The list size is bounded by a very slowly growing function of the block length N ; in particular, it is at most O(log 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. Extended abstracts announcing these results were presented at the 2012 and 2013 ACM Symposia on Theory of Computing (STOC) [16, 17]. This is a merged and revised version of these conference papers, that accounts for the explicit subspace designs that were constructed in [8] subsequent to [17], and makes some simplifications and improvements in the construction of h.s.e sets in Section 6 compared to [16]. The research of V. Guruswami was supported in part by a Packard Fellowship and NSF grants CCF-0963975 and CCF-1422045. Some of this work was done during visits by the author to Nanyang Technological University. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. The research of C. Xing was supported by the Singapore MoE Tier 1 grants RG20/13 and RG25/16, and NTU grant M4081575. 2 V. GURUSWAMI AND C. XING