### On metric regularity of Reed-Muller codes

Alexey Oblaukhov

##### 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 metrically regular set. The problem of investigating metrically regular sets appeared when studying 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.

Available format(s)
Category
Foundations
Publication info
Preprint. MINOR revision.
Keywords
metrically regular setmetric complementcovering radiusbent functionReed-Muller codedeep hole
Contact author(s)
oblaukhov @ gmail com
History
2020-10-19: last of 2 revisions
See all versions
Short URL
https://ia.cr/2019/1481

CC BY

BibTeX

@misc{cryptoeprint:2019/1481,
author = {Alexey Oblaukhov},
title = {On metric regularity of Reed-Muller codes},
howpublished = {Cryptology ePrint Archive, Paper 2019/1481},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/1481}},
url = {https://eprint.iacr.org/2019/1481}
}

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