The Exact Weight Spectrum of RM(m−7,m): A Counterexample to the Carlet-Solé Conjecture
Abstract
Boolean functions are widely used in cryptography and coding theory. A key challenge lies in finding the weight spectrum for all n-variable Boolean functions of degree at most r, also known as the Reed--Muller code RM(r,n). In 2023, Carlet and Solé determined the weight spectra of RM(n−c,n) for c=3,4, and conjectured the weight spectrum of RM(n−c, n) for n≥2c. The weight spectra of RM(n−5, n) and RM(n−6,n) were determined by Carlet in 2024 and by Lou and Wang in 2025, respectively, confirming the Carlet-Solé conjecture for the cases c=5,6. Very recently, Leuenberger et al. and Wei et al. investigated the weight spectrum of RM(7,14) independently, and there are eight potential weights which have not been determined. Up until now, the weight spectra of the Reed--Muller codes RM(r,n) are still unknown for r=3,...,n−7. In this paper, we determine the exact weight spectrum of RM(m−7, m) for every m≥ 14. By combining affine geometry and Fourier analysis, we exclude the conjectured eight weights for RM(7,14). We then extend the exclusions to all higher dimensions using a geometric transfer. Consequently, we provide a counterexample to the Carlet-Solé conjecture.
Related
BibTeX
@misc{lou2026exact,
title = {{The Exact Weight Spectrum of $\RM(m-7,m$): A Counterexample to the Carlet-Solé Conjecture}},
author = {Yueying Lou and Qichun Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36687},
eprint = {2610.36687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36687},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Exact Weight Spectrum of RM(m−7,m): A Counterexample to the Carlet-Solé Conjecture
AU - Yueying Lou
AU - Qichun Wang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36687
UR - https://arxiv.org/abs/2610.36687
AB - Boolean functions are widely used in cryptography and coding theory. A key challenge lies in finding the weight spectrum for all n-variable Boolean functions of degree at most r, also known as the Reed--Muller code RM(r,n). In 2023, Carlet and Solé determined the weight spectra of RM(n−c,n) for c=3,4, and conjectured the weight spectrum of RM(n−c, n) for n≥2c. The weight spectra of RM(n−5, n) and RM(n−6,n) were determined by Carlet in 2024 and by Lou and Wang in 2025, respectively, confirming the Carlet-Solé conjecture for the cases c=5,6. Very recently, Leuenberger et al. and Wei et al. investigated the weight spectrum of RM(7,14) independently, and there are eight potential weights which have not been determined. Up until now, the weight spectra of the Reed--Muller codes RM(r,n) are still unknown for r=3,...,n−7. In this paper, we determine the exact weight spectrum of RM(m−7, m) for every m≥ 14. By combining affine geometry and Fourier analysis, we exclude the conjectured eight weights for RM(7,14). We then extend the exclusions to all higher dimensions using a geometric transfer. Consequently, we provide a counterexample to the Carlet-Solé conjecture.
ER -