The Asymmetric Hamming Bidistance and Distributions over Binary Asymmetric Channels

TL;DR

This paper introduces the asymmetric Hamming bidistance (AHB) to analyze binary asymmetric channels, providing new bounds on error probabilities and detailed code characterizations.

Contribution

It proposes the AHB concept for finer code analysis, derives a new upper bound on error probability, and computes AHB distributions for various code families.

Findings

New AHB measure captures directional discrepancies between codewords.

Derived a novel upper bound on average error probability under maximum-likelihood decoding.

Computed AHB distributions for multiple code families, including projective and nonlinear codes.

Abstract

The binary asymmetric channel is a model for practical communication systems where the error probabilities for symbol transitions $0\rightarrow 1$ and $1\rightarrow0$ differ substantially. In this paper, we introduce the notion of asymmetric Hamming bidistance (AHB) and its two-dimensional distribution, which separately captures directional discrepancies between codewords. This finer characterization enables a more discriminative analysis of decoding the error probabilities for maximum-likelihood decoding (MLD), particularly when conventional measures, such as weight distributions and existing discrepancy-based bounds, fail to distinguish code performance. Building on this concept, we derive a new upper bound on the average error probability for binary codes under MLD and show that, in general, it is incomparable with the two existing bounds derived by Cotardo and Ravagnani (IEEE Trans. Inf. Theory, 68 (5), 2022). To demonstrate its applicability, we compute the complete AHB distributions for several families of codes, including two-weight and three-weight projective codes (with the zero codeword removed) via strongly regular graphs and 3-class association schemes, as well as nonlinear codes constructed from symmetric balanced incomplete block designs (SBIBDs).