{
  "abstract": "Ding \\cite{Ding2016} proposed ten conjectures on binary linear codes arising from Boolean functions. Four of them, namely Conjectures 38--41, were subsequently proved by Göloğlu and Krasnayová \\cite{GologluKrasnayova2019}. In this paper, we investigate the remaining six conjectures, namely Conjectures 19, 27, 30, 33, 34, and 37. For Conjectures~19 and~27, we obtain common weight restrictions and several infinite five-weight families. For Conjecture~30, we prove that every admissible code has three, four, or five nonzero weights, and an explicit four-weight example disproves the original ``three or five weights'' assertion. For Conjecture~33, an infinite five-weight family is obtained. For Conjecture~34, we obtain a general $(2h+1)$-weight upper bound and give an explicit six-weight counterexample, showing that the original ``three or five weights'' assertion is false in general, where $h$ is a positive integer. The case $h=3$ with $3\\nmid m$ is also completely determined. Finally, Conjecture~37 is completely resolved by combining the known results of Ahmadi and Shafaeiabr \\cite{AhmadiShafaeiabr2023} with the treatment of the two remaining classes $(a)$ and $(b)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37093",
  "author_total": 6,
  "authors": [
    {
      "name": "Xiaoqiang Wang"
    },
    {
      "name": "Shiyan Xiong"
    },
    {
      "name": "Mu yuan"
    },
    {
      "name": "Jing Qiu"
    },
    {
      "name": "Dabin Zheng"
    },
    {
      "name": "Jiawei He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/amycxyx2kxfbd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37093",
  "fulltext_url": null,
  "id": "amycxyx2kxfbd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37093",
  "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": "W7214904275",
  "pdf_url": "https://arxiv.org/pdf/2609.37093",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/amycxyx2kxfbd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37093",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37093"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Six Families of Binary Codes Arising from Ding's Conjectures",
  "title_text": "Six Families of Binary Codes Arising from Ding's Conjectures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/amycxyx2kxfbd",
  "venue": "arXiv",
  "versions": []
}