A Revisitation of Low-Rate Bounds on the Reliability Function of Discrete Memoryless Channels for List Decoding

TL;DR

This paper revisits and simplifies the proof of low-rate bounds on the reliability function of discrete memoryless channels, correcting errors and extending results for list decoding and general channels.

Contribution

It provides a clearer, more rigorous proof of existing bounds and extends these bounds to list decoding and broader channel models.

Findings

Simplified proof of zero-rate bounds using Ramsey theory

Correction of an error in Blahut's low-rate bound proof

Extension of bounds to list decoding and general channels

Abstract

We revise the proof of low-rate upper bounds on the reliability function of discrete memoryless channels for ordinary and list-decoding schemes, in particular Berlekamp and Blinovsky's zero-rate bound, as well as Blahut's bound for low rates. The available proofs of the zero-rate bound devised by Berlekamp and Blinovsky are somehow complicated in that they contain in one form or another some cumbersome "non-standard" procedures or computations. Here we follow Blinovsky's idea of using a Ramsey-theoretic result by Komlos, and we complement it with some missing steps to present a proof which is rigorous and easier to inspect. Furthermore, we show how these techniques can be used to fix an error that invalidated the proof of Blahut's low-rate bound, which is here presented in an extended form for list decoding and for general channels.