Explicit list-decodable codes with optimal rate for computationally bounded channels

A stochastic code is a pair of encoding and decoding procedures (Enc, Dec) where Enc : {0, 1}^k × {0, 1}^d → {0, 1}ⁿ. The code is (p,L)-list-decodable against a class C of "channel functions" C : {0, 1}ⁿ → {0, 1}ⁿ if for every message m 2 {0, 1}^k and every channel C 2 C that induces at most pn errors, applying Dec on the "received word" C(Enc(m, S)) produces a list of at most L messages that contain m with high probability over the choice of uniform S {0, 1}^d. Note that both the channel C and the decoding algorithm Dec do not receive the random variable S, when attempting to decode. The rate of a code is R = k/n, and a code is explicit if Enc, Dec run in time poly(n). Guruswami and Smith (J. ACM, to appear), showed that for every constants 0 < p < 1/2 and c > 1 there are Monte-Carlo explicit constructions of stochastic codes with rate R ≥ 1-H(p)-ϵ that are (p,L = poly(1/ϵ))-list decodable for size nc channels. Monte-Carlo, means that the encoding and decoding need to share a public uniformly chosen poly(nc) bit string Y , and the constructed stochastic code is (p,L)-list decodable with high probability over the choice of Y . Guruswami and Smith pose an open problem to give fully explicit (that is not Monte-Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper we resolve this open problem, using a minimal assumption: The existence of poly-Time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97). Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against O(log n)-space online channels. (These are channels that have space O(log n) and are allowed to read the input codeword in one pass). We resolve this open problem. Finally, we consider a tighter notion of explicitness, in which the running time of encoding and list-decoding algorithms does not increase, when increasing the complexity of the channel. We give explicit constructions (with rate approaching 1-H(p) for every p ≤ p0 for some p0 > 0) for channels that are circuits of size 2^n(1/d) and depth d. Here, the running time of encoding and decoding is a fixed polynomial (that does not depend on d). Our approach builds on the machinery developed by Guruswami and Smith, replacing some probabilistic arguments with explicit constructions. We also present a simplified and general approach that makes the reductions in the proof more efficient, so that we can handle weak classes of channels.