Folded Codes from Function Field Towers and Improved Optimal Rate List Decoding
We give a new construction of algebraic codes which are efficiently list decodable from a fraction 1-R- of adversarial errors where R is the rate of the code, for any desired positive constant . The worst-case list size output by the algorithm is O(1/), matching the existential bound for random codes up to constant factors. Further, the alphabet size of the codes is a constant depending only on - it can be made (Õ(1/^2)) which is not much worse than the lower bound of (Ω(1/)). 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 O_(N^c) for an absolute constant c, where N is the code's block length. Our construction is based on a linear-algebraic approach to list decoding folded codes from towers of function fields, and combining it with a special form of subspace-evasive sets. Instantiating this with the explicit "asymptotically good" Garcia-Stichtenoth tower of function fields yields the above parameters. To illustrate the method in a simpler setting, we also present a construction based on Hermitian function fields, which offers similar guarantees with a list and alphabet size polylogarithmic in the block length N. Along the way, we shed light on how to use automorphisms of certain function fields to enable list decoding of the folded version of the associated algebraic-geometric codes.