List Decodability at Small Radii

TL;DR

This paper determines the smallest list size needed for decoding binary error-correcting codes within small radii, providing comprehensive results for various parameters.

Contribution

It explicitly calculates the list decodability parameter $A'(n,d,e)$ for all relevant distances and radii, filling gaps in existing knowledge.

Findings

$A'(n,d,e)$ is known for all $d \\geq 2e-3$.

Complete determination of $A'(n,d,e)$ for all $e \\leq 4$, except 42 cases.

Provides new bounds and exact values for list decodability parameters.

Abstract

$A'(n,d,e)$, the smallest $\ell$ for which every binary error-correcting code of length $n$ and minimum distance $d$ is decodable with a list of size $\ell$ up to radius $e$, is determined for all $d\geq 2e-3$. As a result, $A'(n,d,e)$ is determined for all $e\leq 4$, except for 42 values of $n$.