A Distance Between Channels: the average error of mismatched channels

TL;DR

This paper introduces a new metric for measuring the distance between channels based on their maximum likelihood decoders, providing insights into their decoding similarities and differences.

Contribution

It defines a coding distance between channels grounded in ML-decoder equivalence, supported by explicit probability formulas and a geometric partitioning of channel space.

Findings

Channels with similar ML-decoders are closer in the defined metric.

The space of channels is partitioned into a hyperplane arrangement based on decoder equivalence.

Explicit formulas for the probability that two channels share the same ML-decoder.

Abstract

Two channels are equivalent if their maximum likelihood (ML) decoders coincide for every code. We show that this equivalence relation partitions the space of channels into a generalized hyperplane arrangement. With this, we define a coding distance between channels in terms of their ML-decoders which is meaningful from the decoding point of view, in the sense that the closer two channels are, the larger is the probability of them sharing the same ML-decoder. We give explicit formulas for these probabilities.