List Decoding Bounds for Binary Codes with Noiseless Feedback

TL;DR

This paper investigates the limits of list decoding in binary error-correcting codes with noiseless feedback, providing the first bounds on list decoding radius for different list sizes, including an exact value for size two.

Contribution

It establishes the first nontrivial bounds on list decoding radius for feedback codes, including an exact radius for list size two and upper bounds for larger lists.

Findings

List decoding radius for list size 2 is exactly 3/7.

Upper bounds on list decoding radius for larger lists are provided.

Techniques for list size 2 do not extend to larger list sizes in general.

Abstract

In an error-correcting code, a sender encodes a message $x \in \{ 0, 1 \}^k$ such that it is still decodable by a receiver on the other end of a noisy channel. In the setting of \emph{error-correcting codes with feedback}, after sending each bit, the sender learns what was received at the other end and can tailor future messages accordingly. While the unique decoding radius of feedback codes has long been known to be $\frac13$, the list decoding capabilities of feedback codes is not well understood. In this paper, we provide the first nontrivial bounds on the list decoding radius of feedback codes for lists of size $\ell$. For $\ell = 2$, we fully determine the $2$-list decoding radius to be $\frac37$. For larger values of $\ell$, we show an upper bound of $\frac12 - \frac{1}{2^{\ell + 2} - 2}$, and show that the same techniques for the $\ell = 2$ case cannot match this upper bound in general.