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

We give new constructions of two classes algebraic code families that are efficiently list decodable with small output size from a fraction 1-R-ε adversarial errors, where R is the rate code, for any desired positive constant ε. The alphabet depends only ε and nearly optimal. first class codes obtained by folding algebraic-geometric using automorphisms underlying function field. second restricting evaluation points an to rational subfield . In both cases, we develop linear-algebraic approach perform decoding, which pins down candidate messages subspace nice “periodic” structure. To prune this obtain good bound on size, pick subcodes these pre-coding into certain subspace-evasive sets guaranteed have intersection sort periodic subspaces arise in our decoding. approaches constructing such sets. Monte Carlo construction hierearchical (h.s.e.) leads excellent but not explicit. exploits further ultra-periodicity uses novel construct called designs , were subsequently constructed explicitly also found applications pseudorandomness. get family over fixed instantiate based Garcia–Stichtenoth tower fields. Combining pruning via h.s.e. yields list-decodable up error bounded O (1/ε), matching existential random factors. Further, can be made exp ( Õ (1/ε 2 )), much worse than lower (Ω (1/ε)). parameters achieve thus quite close bounds all three aspects (error-correction radius, size) simultaneously. This is, however, claimed list-decoding property holds high probability. Once (efficiently) sampled, encoding/decoding algorithms deterministic running time _ε N c ) absolute code’s block length. Using instead pruning, efficient decoding errors (Õ(1/ε )). list-size upper very slowly growing length ; particular, it at most O(log r (the th iterated logarithm) integer explicit avoids shortcoming sampling expense slightly size.