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

Abstract

A stochastic code is a pair of encoding and decoding procedures where Encoding procedure receives a k bit message m, and a d bit uniform string S. The code is (p,L)-list-decodable against a class C of "channel functions" from n bits to n bits, if for every message m and every channel C in C that induces at most $pn$ errors, applying decoding 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). Note that both the channel C and the decoding algorithm Dec do not receive the random variable S. The rate of a code is the ratio between the message length and the encoding length, 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)-epsilon that are (p,L=poly(1/epsilon))-list decodable for size n^c channels. Monte-Carlo, means that the encoding and decoding need to share a public uniformly chosen poly(n^c) 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 <= p_0 for some p_0>0) for channels that are circuits of size 2^{n^{Omega(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.