The Metric Structure of Codes for the Binary Symmetric Channel

Abstract : The main result of this paper, theorem 4, is that there exists an infinite sequence of binary digits which can be used to generate sliding parity check codes for the binary symmetric channel (BSC), and the error probability for such codes is close, in a certain sense, to minimum uniformly in all of the parameters: block length, probability of error for a single digit, and rate not too much less than capacity. This result is obtained by studying the metric structure which can be required of codes, and then relating this to the error probability. The paper is essentially self-contained.