The List-Decoding Size of Reed-Muller Codes

TL;DR

This paper provides comprehensive bounds on the list-decoding size and weight distribution of Reed-Muller codes for all distances, extending previous results limited to specific distance ranges.

Contribution

It introduces asymptotic bounds for list-decoding size and weight distribution of Reed-Muller codes applicable at all distances, surpassing prior limitations.

Findings

Bounds on list-decoding size for all distances

Bounds on weight distribution applicable at all distances

Extension of previous results beyond minimum distance

Abstract

In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman \cite{GKZ08} on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this work we provide asymptotic bounds for the list-decoding size of Reed-Muller codes that apply for {\em all} distances. Additionally, we study the weight distribution of Reed-Muller codes. Prior results of Kasami and Tokura \cite{KT70} on the structure of Reed-Muller codewords up to twice the minimum distance, imply bounds on the weight distribution of the code that apply only until twice the minimum distance. We provide accumulative bounds for the weight distribution of Reed-Muller codes that apply to {\em all} distances.