Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels

A stochastic code is a pair of encoding and decodingprocedures (Enc, Dec) where Enc : { 0 , 1 } ^k× { 0 , 1 } ^d→ { 0 , 1 } ⁿ. Thecode is (p, L)-list decodable against a class C of “channel functions”C: { 0 , 1 } ⁿ→ { 0 , 1 } ⁿ if for every message m∈ { 0 , 1 } ^k and every channelC∈ C that induces at most pn errors, applying Dec on the “receivedword” C(Enc(m,S)) produces a list of at most L messages that containm with high probability over the choice of uniform S← { 0 , 1 } ^d. Notethat both the channel C and the decoding algorithm Dec do not receivethe random variable S, when attempting to decode. The rate of a codeis R= k/ n, and a code is explicit if Enc, Dec run in time poly(n). Guruswami and Smith (Journal of the ACM, 2016) showed that forevery constants 0<p<12,ϵ>0 and c> 1 there exist a constantL and a Monte Carlo explicit constructions of stochastic codes withrate R≥ 1 - H(p) - ϵ that are (p, L)-list decodable for size n^c channels.Here, Monte Carlo means that the encoding and decoding need to sharea public uniformly chosen poly (n^c) bit string Y, and the constructedstochastic code is (p, L)-list decodable with high probability over thechoice of Y. Guruswami and Smith pose an open problem to give fully explicit (thatis not Monte Carlo) explicit codes with the same parameters, underhardness assumptions. In this paper, we resolve this open problem,using a minimal assumption: the existence of poly-time computablepseudorandom generators for small circuits, which follows from standardcomplexity assumptions by Impagliazzo and Wigderson (STOC97). Guruswami and Smith also asked to give a fully explicit unconditionalconstructions with the same parameters against O(log n) -space onlinechannels. (These are channels that have space O(log n) and are allowedto read the input codeword in one pass.) We also resolve this openproblem. Finally, we consider a tighter notion of explicitness, in which the runningtime of encoding and list-decoding algorithms does not increase, whenincreasing the complexity of the channel. We give explicit constructions(with rate approaching 1 - H(p) for every p≤ p for some p> 0) forchannels that are circuits of size 2nΩ(1/d) and depth d. Here, the runningtime of encoding and decoding is a polynomial that does not depend onthe depth of the circuit. Our approach builds on the machinery developed by Guruswami andSmith, replacing some probabilistic arguments with explicit constructions.We also present a simplified and general approach that makesthe reductions in the proof more efficient, so that we can handle weakclasses of channels.