The Covering Radius of the Reed-Muller Code $RM(m-4,m)$ in $RM(m-3,m)$

TL;DR

This paper introduces methods to compute the distance between Boolean polynomials and lower-degree polynomial spaces within Reed-Muller codes, providing bounds and certificates for the covering radius in specific cases.

Contribution

The paper develops verifiable methods for bounding the covering radius of Reed-Muller codes, with concrete results for specific code parameters and improved bounds for higher degrees.

Findings

Covering radius of RM(4,8) in RM(5,8) is 26

Covering radius of RM(5,9) in RM(6,9) is between 28 and 32

Improved bounds on distances for higher m and for 2-resilient polynomials

Abstract

We present methods for computing the distance from a Boolean polynomial on $m$ variables of degree $m-3$ (i.e., a member of the Reed-Muller code $RM(m-3,m)$) to the space of lower-degree polynomials ($RM(m-4,m)$). The methods give verifiable certificates for both the lower and upper bounds on this distance. By applying these methods to representative lists of polynomials, we show that the covering radius of $RM(4,8)$ in $RM(5,8)$ is 26 and the covering radius of $RM(5,9)$ in $RM(6,9)$ is between 28 and 32 inclusive, and we get improved lower bounds for higher~$m$. We also apply our methods to various polynomials in the literature, thereby improving the known bounds on the distance from 2-resilient polynomials to $RM(m-4,m)$.