On Equivalence of Binary Asymmetric Channels regarding the Maximum Likelihood Decoding

We study the problem of characterizing when two memoryless binary asymmetric channels, described by their transition probabilities $(p,q)$ and $(p^{\mathrm{\prime }},q^{\mathrm{\prime }})$, are equivalent from the point of view of maximum likelihood decoding (MLD) when restricted to $n$-block binary codes. This equivalence of channels induces a partition (depending on $n$) on the space of parameters $(p,q)$ into regions associated with the equivalence classes. Explicit expressions for describing these regions, their number and areas are derived. Some perspectives of applications of our results to decoding problems are also presented.