Weight spectra of some families of GRM codes

2026-09-11

AI Summary

该研究通过归纳法和特定工具计算，完全确定了几类广义Reed-Muller码的重量谱。

Abstract

Let $\mathrm{RM}_q(r,m)$ denote the generalized Reed--Muller code of order $r$ and length $q^m$ over the finite field $\mathbb{F}_q$. We completely determine the weight spectra of $\mathrm{RM}_3(2m-i,m)$ for $i=3,4$, $\mathrm{RM}_4(3m-i,m)$ for $i=2,3$, $\mathrm{RM}_5(4m-4,m)$, and $\mathrm{RM}_7(6m-4,m)$, in the ranges of $m$ specified in the corresponding results. The proofs proceed by induction on $m$, using a sumset inclusion for GRM weight spectra together with results on low-weight codewords. The required base cases are established by computations in Magma and, for certain ternary cases, exact constraint solving with Z3.

Problem

• weight spectra
• generalized Reed--Muller codes
• finite field

Innovation

• weight spectra
• generalized Reed--Muller codes
• induction on m
• sumset inclusion
• low-weight codewords