The Sphere Packing Bound For Memoryless Channels

TL;DR

This paper derives tight sphere packing bounds with polynomial prefactors for certain classes of memoryless channels, providing insights into the fundamental limits of coding performance and error decay rates.

Contribution

It introduces new sphere packing bounds for non-stationary and stationary memoryless channels with convex input constraints, extending classical results.

Findings

Derived sphere packing bounds with polynomial prefactors

Established tightness of bounds in error probability decay

Applied Augustin's and Gallager's methods to new channel models

Abstract

Sphere packing bounds (SPBs) ---with prefactors that are polynomial in the block length--- are derived for codes on two families of memoryless channels using Augustin's method: (possibly non-stationary) memoryless channels with (possibly multiple) additive cost constraints and stationary memoryless channels with convex constraints on the composition (i.e. empirical distribution, type) of the input codewords. A variant of Gallager's bound is derived in order to show that these sphere packing bounds are tight in terms of the exponential decay rate of the error probability with the block length under mild hypotheses.