An entropy inequality for q-ary random variables and its application to channel polarization

TL;DR

This paper introduces an entropy inequality for q-ary variables with prime q, enabling a straightforward proof of channel polarization, and demonstrates how to construct channels with capacities that satisfy specific inequalities.

Contribution

It establishes a new entropy inequality for q-ary variables and applies it to prove channel polarization in the q-ary setting.

Findings

Channels W^- and W^+ have capacities satisfying I(W^-)<I(W)<I(W^+)

The entropy inequality simplifies the proof of channel polarization for prime q

The method applies to q-ary channels with prime q

Abstract

It is shown that given two copies of a q-ary input channel $W$, where q is prime, it is possible to create two channels $W^-$ and $W^+$ whose symmetric capacities satisfy $I(W^-)\le I(W)\le I(W^+)$, where the inequalities are strict except in trivial cases. This leads to a simple proof of channel polarization in the q-ary case.