Codes for Computationally Simple Channels: Explicit Constructions with Optimal Rate

Abstract

In this paper, we consider coding schemes for computationally bounded channels, which can introduce an arbitrary set of errors as long as (a) the fraction of errors is bounded with high probability by a parameter p and (b) the process which adds the errors can be described by a sufficiently "simple" circuit. Codes for such channel models are attractive since, like codes for standard adversarial errors, they can handle channels whose true behavior is unknown or varying over time. For three classes of channels, we provide explicit, efficiently encodable/decodable codes of optimal rate where only inefficiently decodable codes were previously known. In each case, we provide one encoder/decoder that works for every channel in the class. Unique decoding for additive errors: We give the first construction of a poly-time encodable/decodable code for additive (a.k.a. oblivious) channels that achieve the Shannon capacity 1-H(p). List-decoding for online log-space channels: A space-S(N) bounded channel reads and modifies the transmitted codeword as a stream, using at most S(N) bits of workspace on transmissions of N bits. For constant S, this captures many models from the literature, including "discrete channels with finite memory" and "arbitrarily varying channels". We give an efficient code with optimal rate (arbitrarily close to 1-H(p)) that recovers a short list containing the correct message with high probability for channels which read and modify the transmitted codeword as a stream, using at most O(\log N) bits of workspace on transmissions of N bits. List-decoding for poly-time channels: For any constant c we give a similar list-decoding result for channels describable by circuits of size at most N^c, assuming the existence of pseudorandom generators.