On the Symmetries of the Deletion Channel
Abstract
In this paper, we consider a class of symmetry groups associated to communication channels, which can informally be viewed as the transformations of the set of inputs that ``commute'' with the action of the channel. These groups were first studied by Polyanskiy in (IEEEToIT 2013). We show the simple result that the input distribution that attains the maximum mutual information for a given channel is a ``fixed point'' of its group. We conjecture (and give empirical evidence) that the channel group of the deletion channel is extremely small (it contains a number of elements constant in the blocklength). We prove a special case of this conjecture. This serves as some formal justification for why the analysis of the binary deletion channel has proved much more difficult than its memoryless counterparts.
Keywords
Cite
@article{arxiv.2202.03633,
title = {On the Symmetries of the Deletion Channel},
author = {Francisco Pernice},
journal= {arXiv preprint arXiv:2202.03633},
year = {2022}
}
Comments
V2: Substantial revision. Polyanskiy's work, an important reference missing in the prior version, is now cited. A theorem has been removed due to a bug in the proof (see Correction section). We thank anonymous reviewers whose comments led to these changes. A special case of the conjecture in the previous version is also now proved. V1: ancillary files included