# The Augustin Center and The Sphere Packing Bound For Memoryless Channels

## TL;DR

This paper establishes the existence and uniqueness of the Augustin center and bounds for channels with convex constraints, introduces related capacities and radii, and derives sphere packing bounds for various memoryless channels.

## Contribution

It introduces Augustin-Legendre capacity, center, and radius, proving their equivalence to Renyi-Gallager entities, and derives sphere packing bounds with polynomial prefactors for specific channel families.

## Key findings

- Existence of a unique Augustin center for channels with convex constraints.
- Equivalence of Augustin-Legendre and Renyi-Gallager capacities and centers.
- Sphere packing bounds with polynomial prefactors for certain memoryless channels.

## Abstract

For any channel with a convex constraint set and finite Augustin capacity, existence of a unique Augustin center and associated Erven-Harremoes bound are established. Augustin-Legendre capacity, center, and radius are introduced and proved to be equal to the corresponding Renyi-Gallager entities. Sphere packing bounds with polynomial prefactors are derived for codes on two families of channels: (possibly non-stationary) memoryless channels with multiple additive cost constraints and stationary memoryless channels with convex constraints on the empirical distribution of the input codewords.