Parameters of Codes for the Binary Asymmetric Channel

TL;DR

This paper introduces new parameters for binary error-correcting codes tailored to the binary asymmetric channel, providing bounds and examples that enhance understanding of code performance in this context.

Contribution

The paper defines two novel discrepancy-based parameters for binary codes that relate to decoding failure probabilities and derives bounds involving these parameters.

Findings

New discrepancy measures for binary vectors

Bounds on code size and weight distribution using these parameters

Examples of codes achieving the bounds

Abstract

We introduce two notions of discrepancy between binary vectors, which are not metric functions in general but nonetheless capture the mathematical structure of the binary asymmetric channel. In turn, these lead to two new fundamental parameters of binary error-correcting codes, both of which measure the probability that the maximum likelihood decoder fails. We then derive various bounds for the cardinality and weight distribution of a binary code in terms of these new parameters, giving examples of codes meeting the bounds with equality.