{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13653",
  "author_total": 3,
  "authors": [
    {
      "name": "Minjia Shi"
    },
    {
      "name": "Zhaokang Xing"
    },
    {
      "name": "Patrick Sole"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1dghc4tgh92s6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13653",
  "fulltext_url": null,
  "id": "1dghc4tgh92s6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13653",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.13653",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1dghc4tgh92s6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13653",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13653"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Weight spectra of some families of GRM codes",
  "title_text": "Weight spectra of some families of GRM codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/1dghc4tgh92s6",
  "venue": "arXiv",
  "versions": []
}