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. The encoders are randomized, and probabilities are taken over the (local, unknown to the decoder) coins of the encoder and those of the channel. 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). These are channels which add an arbitrary error vector e ∈ {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. Such channels capture binary symmetric errors and burst errors as special cases. 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 limited to logarithmic space. 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. We are not aware of any channel models considered in the information theory literature, other than purely adversarial channels, which require more than linear-size circuits to implement.
Subsequently, REF further removes the public-key setup assumption specifically for the channel in which either the error is independent of the actual message being sent, or the errors can be described by polynomial size circuits.
Optimal-Rate Code Constructions for Computationally Simple Channels
arXiv: Information Theory
Computer Science Mathematics