Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$
Abstract
We study the set of weights occurring in the high-order binary Reed-Muller codes $RM(m-7,m)$. For every m≥ 14, we prove that the weight spectrum contains all even weights in [336,2m−336], together with an explicit finite set of low weights and their complements. The proof first establishes the corresponding base-code inclusion in RM(7,14). This base step uses four-block concatenation, explicit degree-six monomial witnesses, and known spectrum inputs for RM(5,10) and RM(6,12). The base inclusion is then lifted to all m≥ 14 by a concatenation-based induction. We also analyze the remaining base-code target weights $322,326,330,334$ in several restricted families, including four-block branches, affine-flat and structured inner corrections in a fixed Carlet-type Maiorana-McFarland outer shell, and sparse sums of degree-7 coordinate monomials. The residual analysis delineates several construction routes that cannot yield the four unresolved base-code weights, thereby sharpening the remaining c=7 spectrum problem.
Related
- Common zeros of homogeneous polynomials and the weight hierarchy of projective Reed-Muller codes
- Reed-Muller Codes are Optimal Binary Linear 3-Query LCCs
- Determination of weight enumerators of binary extremal self-dual codes of length 60
- Linear Equivalence of Full Weight Spectrum Cyclic Orbit Codes
- A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes
BibTeX
@misc{wei2026some,
title = {{Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$}},
author = {Zibo Wei and Bo Zhou and Xiao Zhang and Hao Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29807},
eprint = {2610.29807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29807},
}
RIS
TY - UNPB
TI - Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$
AU - Zibo Wei
AU - Bo Zhou
AU - Xiao Zhang
AU - Hao Chen
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29807
UR - https://arxiv.org/abs/2610.29807
AB - We study the set of weights occurring in the high-order binary Reed-Muller codes $RM(m-7,m)$. For every m≥ 14, we prove that the weight spectrum contains all even weights in [336,2^m−336], together with an explicit finite set of low weights and their complements. The proof first establishes the corresponding base-code inclusion in RM(7,14). This base step uses four-block concatenation, explicit degree-six monomial witnesses, and known spectrum inputs for RM(5,10) and RM(6,12). The base inclusion is then lifted to all m≥ 14 by a concatenation-based induction. We also analyze the remaining base-code target weights $322,326,330,334$ in several restricted families, including four-block branches, affine-flat and structured inner corrections in a fixed Carlet-type Maiorana-McFarland outer shell, and sparse sums of degree-7 coordinate monomials. The residual analysis delineates several construction routes that cannot yield the four unresolved base-code weights, thereby sharpening the remaining c=7 spectrum problem.
ER -