{
  "abstract": "A $4$-general set of PG(n,q) is a point set with no four coplanar,and Mb{n}{q} is the largest such size. The climb problem asks, at each $n$, whether the constructions of Pavese (2025) can be improved by one point. We show that the answer is governed by the code tables: $PG(M-1,q)$ contains an $N$-point $4$-general set exactly when a projective $[N,N-M,\\ge 5]_q$ code exists. Over $F_4$ this settles the first two rungs: Mb{3}{4}=5 and Mb{4}{4}=11, each with two independent proofs, so Pavese's constructions are optimal at both. Over $F_5$ the same reduction gives Mb{4}{5}=12, one more than the near-MDS lower bound. At $n=5$ the tables fall silent; we state the verified position $21\\le Mb{5}{4}\\le 30$ rather than the tabulated $\\le 29$, which rests on a private communication. Any $22$-point $4$-general set of $PG(5,4)$, if one exists, shares at most $12$ points with Pavese's extremal $21$-point set; the proof of this rigidity statement is a finite computation, with certificates in Appendix~A.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03272",
  "author_total": 4,
  "authors": [
    {
      "name": "Liangdong Lu"
    },
    {
      "name": "Ruipan Yang"
    },
    {
      "name": "Qiang Fu"
    },
    {
      "name": "Hao Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/28vfb99wxhfn0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03272",
  "fulltext_url": null,
  "id": "28vfb99wxhfn0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03272",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7220473659",
  "pdf_url": "https://arxiv.org/pdf/2610.03272",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/28vfb99wxhfn0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03272",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03272"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The climb problem for $4$-general sets in PG(n,4)",
  "title_text": "The climb problem for $4$-general sets in PG(n,4)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/28vfb99wxhfn0",
  "venue": "arXiv",
  "versions": []
}