On metric regularity of Reed-Muller codes

TL;DR

This paper investigates the metric properties of Reed-Muller codes, establishing their metric regularity for many cases and conjecturing it holds for all such codes, which has implications in cryptography and coding theory.

Contribution

The work describes metric complements and proves metric regularity for specific Reed-Muller codes, extending previous results and supporting the conjecture for all Reed-Muller codes.

Findings

Proved metric regularity of RM(0,m) and RM(k,m) for k e0 m-3.

Established metric regularity of RM(1,5) and RM(2,6).

Supports the conjecture that all Reed-Muller codes are metrically regular.

Abstract

In this work we study metric properties of the well-known family of binary Reed-Muller codes. Let $A$ be an arbitrary subset of the Boolean cube, and $\widehat{A}$ be the metric complement of $A$ -- the set of all vectors of the Boolean cube at the maximal possible distance from $A$. If the metric complement of $\widehat{A}$ coincides with $A$, then the set $A$ is called a {\it metrically regular set}. The problem of investigating metrically regular sets appeared when studying {\it bent functions}, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this work we describe metric complements and establish the metric regularity of the codes $\mathcal{RM}(0,m)$ and $\mathcal{RM}(k,m)$ for $k \geqslant m-3$. Additionally, the metric regularity of the codes $\mathcal{RM}(1,5)$ and $\mathcal{RM}(2,6)$ is proved. Combined with previous results by Tokareva N. (2012) concerning duality of affine and bent functions, this establishes the metric regularity of most Reed-Muller codes with known covering radius. It is conjectured that all Reed-Muller codes are metrically regular.