An Upper Bound on the Reliability Function of the DMC

TL;DR

This paper introduces a new upper bound on the reliability function for discrete memoryless channels, improving upon existing bounds and applicable across various rates and decoding metrics, with practical approximations and analysis for specific channels.

Contribution

The paper presents a novel, unified upper bound on the reliability function that surpasses traditional bounds and applies broadly to different rates and additive decoding metrics.

Findings

The new bound is at least as tight as existing bounds across all rates.

It is particularly tighter at low rates compared to traditional bounds.

A simple approximation function for the reliability function is proposed.

Abstract

We derive a new upper bound on the reliability function for channel coding over discrete memoryless channels. Our bounding technique relies on two main elements: (i) adding an auxiliary genie-receiver that reveals to the original receiver a list of codewords including the transmitted one, which satisfy a certain type property, and (ii) partitioning (most of) the list into subsets of codewords that satisfy a certain pairwise-symmetry property, which facilitates lower bounding of the average error probability by the pairwise error probability within a subset. We compare the obtained bound to the Shannon-Gallager-Berlekamp straight-line bound, the sphere-packing bound, and an amended version of Blahut's bound. Our bound is shown to be at least as tight for all rates, with cases of stricter tightness in a certain range of low rates, compared to all three aforementioned bounds. Our derivation is performed in a unified manner which is valid for any rate, as well as for a wide class of additive decoding metrics, whenever the corresponding zero-error capacity is zero. We further present a relatively simple function that may be regarded as an approximation to the reliability function in some cases. We also present a dual form of the bound, and discuss a looser bound of a simpler form, which is analyzed for the case of the binary symmetric channel with maximum likelihood decoding.