Determining the Weight Spectrum of the Reed--Muller Codes RM(m-6,m)

TL;DR

This paper determines the weight spectrum of Reed-Muller codes RM(m-6,m) for m≥12 using a new construction method, addressing an open problem and proposing a conjecture for further generalization.

Contribution

It introduces a novel construction approach to determine the weight spectrum of RM(m-6,m) for m≥12, advancing understanding of Reed-Muller codes.

Findings

Determined the weight spectrum of RM(m-6,m) for m≥12.

Proposed a conjecture that, if true, would fully solve the open problem.

Verified the conjecture in some specific cases.

Abstract

The weight spectra of the Reed-Muller codes $RM(r,m)$ were unknown for $r=3,...,m-5$. In IEEE Trans. Inform. Theory 2024, Carlet determined the weight spectrum of $RM(m-5,m)$ for $m\ge10$ using the Maiorana-McFarland construction, where the result was tried to be extended to $RM(m-6,m)$, but many problems occurred and much work needed to be done. In this paper, we propose a novel way of constructing Reed--Muller codewords and determine the weight spectrum of $RM(m-6,m)$ for $m\ge12$, which gives a positive answer to an open question on the weight spectrum of $RM(m-c,m)$ for $c=6$. Moreover, we put forward a conjecture and verify it for some cases. If the conjecture is true, then that open question can be completely solved.