Distance distribution of binary codes and the error probability of decoding

TL;DR

This paper derives bounds on the error probability of binary codes over symmetric channels, showing the random coding exponent is optimal in certain rate intervals, with implications for both binary and Gaussian channels.

Contribution

It provides improved bounds on the error probability and establishes the optimality of the random coding exponent in specific rate regions for binary and Gaussian channels.

Findings

The random coding exponent is the true reliability function below the critical rate.

An improved upper bound for the error probability exponent is established.

Results apply to both binary symmetric and Gaussian channels.

Abstract

We address the problem of bounding below the probability of error under maximum likelihood decoding of a binary code with a known distance distribution used on a binary symmetric channel. An improved upper bound is given for the maximum attainable exponent of this probability (the reliability function of the channel). In particular, we prove that the ``random coding exponent'' is the true value of the channel reliability for code rate $R$ in some interval immediately below the critical rate of the channel. An analogous result is obtained for the Gaussian channel.