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).

翻译：二元非对称信道是实际通信系统的模型，其中符号转移$0\rightarrow 1$和$1\rightarrow0$的差错概率存在显著差异。本文提出非对称汉明双距离（AHB）的概念及其二维分布，该概念分别捕捉码字之间的方向性差异。这种更精细的特征化使得对最大似然译码（MLD）的差错概率进行更具区分性的分析成为可能，尤其是在传统度量（如重量分布和现有基于差异的界）无法区分码性能的情况下。基于这一概念，我们推导了MLD下二元码平均差错概率的一个新上界，并证明该界通常与Cotardo和Ravagnani（IEEE Trans. Inf. Theory, 68 (5), 2022）推导的两个现有界不可比较。为展示其适用性，我们计算了若干码族的完整AHB分布，包括通过强正则图和3类结合方案得到的双重量与三重量投影码（移除零码字），以及由对称平衡不完全区组设计（SBIBDs）构造的非线性码。