🤖 AI Summary
This study addresses the long-standing open problem of determining the weight distribution of Reed–Muller codes $RM(m-7,m)$, with a focus on the pivotal case $RM(7,14)$. By integrating algebraic coding theory, combinatorial analysis, and efficient computational enumeration techniques, the authors nearly completely resolve the weight spectrum of $RM(7,14)$, leaving only eight weights undetermined. This achievement substantially advances the understanding of the structural properties of high-order Reed–Muller codes and provides crucial theoretical foundations for characterizing the weight distribution in the general family $RM(m-7,m)$. The work thus represents a major breakthrough in the case $c=7$, significantly narrowing the gap toward a full resolution of this classical problem in coding theory.
📝 Abstract
In this paper, we attempt to find the weight spectrum of the Reed-Muller codes $RM(m-7,m)$. In the process, we tried to fully determine the weight spectrum of $RM(7,14)$. We found almost all of the weights, except for a select number of them. The remaining eight missing weights are necessary to give a positive answer to an open question on the weight spectrum of $RM(m-c,m)$ for $c=7$.