A mutual information invariance approach to symmetry in discrete memoryless channels

Abstract

There are numerous notions of symmetry for discrete memoryless channels. A common goal of these various definitions is that the capacity may be easily computed once the channel is declared to be symmetric. In this paper we focus on a class of definitions of symmetry characterized by the invariance of the channel mutual information over a group of permutations of the input distribution. For definitions of symmetry within this class, we give a simple proof of the optimality of the uniform distribution. The fundamental channels are all symmetric with a general enough definition of symmetry. This paper provides a definition of symmetry that covers these fundamental channels along with a proof that is simple enough to find itself on the chalkboard of even the most introductory class in information theory.