List-Decoding Capacity Implies Capacity on the q-ary Symmetric Channel

TL;DR

This paper establishes a formal connection between list-decoding capacity and the Shannon capacity of the q-ary symmetric channel, showing that certain codes achieving list-decoding capacity also achieve channel capacity.

Contribution

It proves that linear codes with sufficiently large minimum distance that attain list-decoding capacity also achieve the Shannon capacity on the q-ary symmetric channel.

Findings

Linear codes with minimum distance _{ ext{min}}=\u2208(q^3) achieve capacity on the q-ary symmetric channel.

List-decoding capacity and channel capacity are formally connected for these codes.

The result provides a black-box link between adversarial list-decoding and stochastic channel capacity.

Abstract

It is known that the Shannon capacity of the q-ary symmetric channel (qSC) is the same as the list-decoding capacity of an adversarial channel, raising the question of whether there is a formal (and black-box) connection between the two. We show that there is: Any linear code $C\subseteq \mathbb{F}_q^n$ that has minimum distance $d_{\min}=\omega(q^3)$ and achieves list-decoding capacity also achieves capacity on the qSC.