Optimal-Rate Code Constructions for Computationally Simple Channels

Abstract: 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 two 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. The encoders are randomized, and probabilities are taken over the (local, unknown to the decoder) coins of the encoder and those of the channel. (1) Unique decoding for additive errors: We give the first construction of a polynomial-time encodable/decodable code for additive (a.k.a. oblivious) channels that achieve the Shannon capacity $1-H(p)$. These channels add an arbitrary error vector $e\in{0,1}^N$ of weight at most $pN$ to the transmitted word; the vector $e$ can depend on the code but not on the particular transmitted word. (2) List-decoding for polynomial-time channels: For every constant $c>0$, we give a Monte Carlo construction of an code with optimal rate (arbitrarily close to $1-H(p)$) that efficiently recovers a short list containing the correct message with high probability for channels describable by circuits of size at most $N^c$. We justify the relaxation to list-decoding by showing that even with bounded channels, uniquely decodable codes cannot have positive rate for $p>1/4$.