The Sphere Packing Bound via Augustin's Method

TL;DR

This paper derives a polynomial prefactor sphere packing bound for product channels with a maximum Renyi capacity growing logarithmically with block length, providing new bounds on reliability functions with feedback.

Contribution

It introduces a sphere packing bound with polynomial prefactor for channels with certain Renyi capacity growth, extending previous asymptotic results to non-asymptotic and milder stationarity cases.

Findings

Established a non-asymptotic sphere packing bound with polynomial prefactor.

Bound the reliability function of stationary product channels with feedback.

Extended the sphere packing bound to cases with milder stationarity assumptions.

Abstract

A sphere packing bound (SPB) with a prefactor that is polynomial in the block length $n$ is established for codes on a length $n$ product channel $W_{[1,n]}$ assuming that the maximum order $1/2$ Renyi capacity among the component channels, i.e. $\max_{t\in[1,n]} C_{1/2,W_{t}}$, is $\mathit{O}(\ln n)$. The reliability function of the discrete stationary product channels with feedback is bounded from above by the sphere packing exponent. Both results are proved by first establishing a non-asymptotic SPB. The latter result continues to hold under a milder stationarity hypothesis.