On the bit error rate of repeated error-correcting codes

Abstract

Classically, error-correcting codes are studied with respect to performance metrics such as minimum distance (combinatorial) or probability of bit/block error over a given stochastic channel. In this paper, a different metric is considered. It is assumed that the block code is used to repeatedly encode user data. The resulting stream is subject to adversarial noise of given power, and the decoder is required to reproduce the data with minimal possible bit-error rate. This setup may be viewed as a combinatorial joint source-channel coding. Two basic results are shown for the achievable noise-distortion tradeoff: the optimal performance for decoders that are informed of the noise power, and global bounds for decoders operating in complete oblivion (with respect to noise level). General results are applied to the Hamming [7, 4, 3] code, for which it is demonstrated (among other things) that no oblivious decoder exist that attains optimality for all noise levels simultaneously.