{
  "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 $n\\ge2c$. 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 $m\\ge 14$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36687",
  "author_total": 2,
  "authors": [
    {
      "name": "Yueying Lou"
    },
    {
      "name": "Qichun Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dd5z7ez53w8xz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36687",
  "fulltext_url": null,
  "id": "dd5z7ez53w8xz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36687",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7224122019",
  "pdf_url": "https://arxiv.org/pdf/2610.36687",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dd5z7ez53w8xz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36687",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36687"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Exact Weight Spectrum of $\\RM(m-7,m$): A Counterexample to the Carlet-Solé Conjecture",
  "title_text": "The Exact Weight Spectrum of RM(m−7,m): A Counterexample to the Carlet-Solé Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/dd5z7ez53w8xz",
  "venue": "arXiv",
  "versions": []
}