The Covering Radius of the Reed--Muller Code $RM(2,7)$ is 40

TL;DR

This paper determines the exact covering radius of the Reed--Muller code RM(2,7) as 40, resolving a long-standing open problem and providing new bounds for higher dimensions.

Contribution

It proves the covering radius of RM(2,7) is 40, matching that in RM(3,7), and introduces new upper bounds for RM(2,n) for n=8,9,10.

Findings

Covering radius of RM(2,7) is 40.

Equivalent covering radius in RM(3,7).

New upper bounds for RM(2,n) for n=8,9,10.

Abstract

It was proved by J. Schatz that the covering radius of the second order Reed--Muller code $RM(2, 6)$ is 18 (IEEE Trans Inf Theory 27: 529--530, 1985). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for $RM(2,n)$, $n=8,9,10$.