{
  "abstract": "We study the set of weights occurring in the high-order binary Reed-Muller codes $RM(m-7,m)$. For every $m\\geq 14$, 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 $m\\geq 14$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29807",
  "author_total": 4,
  "authors": [
    {
      "name": "Zibo Wei"
    },
    {
      "name": "Bo Zhou"
    },
    {
      "name": "Xiao Zhang"
    },
    {
      "name": "Hao Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40s978tyyggwq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29807",
  "fulltext_url": null,
  "id": "40s978tyyggwq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29807",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223240895",
  "pdf_url": "https://arxiv.org/pdf/2610.29807",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40s978tyyggwq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29807",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29807"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$",
  "title_text": "Some Results on the Weight Spectrum of Binary Reed-Muller Codes $RM(m-7,m)$",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/40s978tyyggwq",
  "venue": "arXiv",
  "versions": []
}