Channel Metrization

TL;DR

This paper introduces an algorithm to identify if a channel can be associated with a metric where maximum likelihood and minimum distance decoding are equivalent, and demonstrates the universality of the Hamming metric for decoding.

Contribution

It provides a method to determine compatible metrics for channels and proves the Hamming metric's universality in decoding scenarios.

Findings

Algorithm for channel metric compatibility

Embedding of metrics into hypercube with Hamming metric

Hamming metric's universality in decoding

Abstract

We present an algorithm that, given a channel, determines if there is a distance for it such that the maximum likelihood decoder coincides with the minimum distance decoder. We also show that any metric, up to a decoding equivalence, can be isometrically embedded into the hypercube with the Hamming metric, and thus, in terms of decoding, the Hamming metric is universal.