The Asymmetric Hamming Bidistance and Distributions over Binary Asymmetric Channels

2026-04-08

AI Summary

This work addresses the challenge of accurately characterizing error performance under maximum-likelihood decoding in binary asymmetric channels, where the transition probabilities of 0→1 and 1→0 differ significantly—a scenario poorly captured by conventional coding-theoretic analyses. The authors introduce the asymmetric Hamming bidirectional distance (AHB) and its two-dimensional distribution to precisely model pairwise codeword asymmetry. Leveraging this framework, they derive a novel upper bound on the average error probability that strictly improves upon existing bounds and is incomparable with them. Using tools from combinatorial design theory—including strongly regular graphs, three-class association schemes, and symmetric balanced incomplete block designs (SBIBDs)—they construct several families of nonlinear codes and, for the first time, fully compute the AHB distributions for binary and ternary projective codes (excluding the all-zero codeword) as well as SBIBD-based codes, thereby substantially enhancing the accuracy of performance analysis in asymmetric channels.

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).

Problem

• binary asymmetric channel
• error probability
• maximum-likelihood decoding
• code performance
• asymmetric errors

Innovation

• asymmetric Hamming bidistance
• binary asymmetric channel
• maximum-likelihood decoding
• error probability bound
• code distribution