Paper 2026/2623

The Covering Radius of $\operatorname{RM}(2,8)$ is Between $88$ and $91$

Abstract

Let $\operatorname{RM}(r,n)$ be the $r$th order Reed--Muller code of length $2^n$, and let $\rho(r,n)$ denote its covering radius. The values $\rho(2,6)=18$ and $\rho(2,7)=40$ are known, whereas the exact value of $\rho(2,8)$ remains open. Starting from the current state of the art bound $88\leq\rho(2,8)\leq96$, we give a computer-assisted reduction of the upper bound $\rho(2,8)\leq91$. We first show that every even-weight Boolean function on eight variables is within distance $90$ of a quadratic function. Changing one value then gives the bound $91$ for odd-weight functions. Our main lemma addresses rank-four seven-variable quintics: when their second-order distance is $36$ and their maximum quadratic-shift first-order distance is $56$, they have at least $60$ minimizing quadratic cosets and at most $12$ cosets of cost $56$. The complete proof covers all $68443$ quartic classes. For every rank-four quintic, the distance-$56$ cosets are enumerated outright, and the $5199$ words are examined individually. Profile counts, hyperplane contractions, and support moments then give the even-weight bound. We describe the calculations and enumerations, together with the classification upon which the proof depends. Instructions to verify the computer search portion may be found at $\texttt{https://github.com/EvelynSabo/RM28}$.