{
  "abstract": "The generalized packing--covering conjecture of Elimelech, Firer and Schwartz asserts that, for every linear code $\\mathcal{C}$ and every admissible order $t$, the $t$-th generalized Hamming weight $d_t(\\mathcal{C})$ and the $t$-th generalized covering radius $R_t(\\mathcal{C})$ satisfy $d_t(\\mathcal{C})\\le 2R_t(\\mathcal{C})+2$. We give a computer-assisted proof of the conjecture for every linear code over every finite field and every admissible order. Combining a parity-check reformulation of the conjecture, bounds on the length of putative counterexamples, and successive puncturing arguments, we settle all orders $t\\ge 32$ and reduce the remaining orders to finitely many parameter tuples, which we exclude by an exact computer verification.",
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  "article_type": null,
  "arxiv": "2609.34910",
  "author_total": 4,
  "authors": [
    {
      "name": "Gianira N. Alfarano"
    },
    {
      "name": "Giuseppe Marino"
    },
    {
      "name": "Alessandro Neri"
    },
    {
      "name": "Rocco Trombetti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjphw3evnhkfe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34910",
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  "pdf_url": "https://hal.science/hal-05772903/document",
  "pmcid": null,
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  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjphw3evnhkfe.ris",
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      "first_seen": "2026-10-30T12:17:45Z",
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      "url": "https://hal.science/hal-05772903v1"
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      "native_id": "2609.34910",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34910"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
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  ],
  "title": "A proof of the generalized packing-covering conjecture",
  "title_text": "A proof of the generalized packing-covering conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/cjphw3evnhkfe",
  "venue": "HAL",
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}