{
  "abstract": "Chen and Xie recently proved, using the Ashikhmin--Barg lemma on minimal vectors, that every binary linear $[n,k,d]$ code with $k=n-2d+2+v$ ($v\\ge 0$) has no codewords of weight in the interval $[2d-v,\\,2d-1]$. Their argument uses two of the five basic properties of minimal vectors established by Ashikhmin and Barg (1998). In this note we utilize the third property, the disjoint-support decomposition of non-minimal codewords in binary codes, to generate a \\emph{mirror} vanishing band on the other side of $2d$: if $A_{d+1}=\\cdots=A_{d+t}=0$ for some $t\\ge 1$ and $k\\ge n-2d+1$, then $A_w=0$ for all $w\\in[2d+1,\\,2d+t]$. Combining the two bands, the number of nonzero weights of such a code is at most $n-d-v-2t+1$, improving the Chen--Xie bound $n-d+1-v$ by $2t$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20344",
  "author_total": 1,
  "authors": [
    {
      "name": "Xianmang He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0jrkabwr42tk6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20344",
  "fulltext_url": null,
  "id": "0jrkabwr42tk6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20344",
  "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": "W7213576962",
  "pdf_url": "https://arxiv.org/pdf/2609.20344",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0jrkabwr42tk6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20344",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20344"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes",
  "title_text": "A Mirror Vanishing Band for Weight Distributions of Binary Linear Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/0jrkabwr42tk6",
  "venue": "arXiv",
  "versions": []
}