The climb problem for $4$-general sets in PG(n,4)

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,≥ 5]_q code exists. Over F₄ 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 F₅ 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≤ Mb54≤ 30 rather than the tabulated ≤ 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.

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BibTeX RIS

BibTeX

@misc{lu2026climb,
  title = {{The climb problem for $4$-general sets in PG(n,4)}},
  author = {Liangdong Lu and Ruipan Yang and Qiang Fu and Hao Song},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03272},
  eprint = {2610.03272},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03272},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The climb problem for $4$-general sets in PG(n,4)
AU  - Liangdong Lu
AU  - Ruipan Yang
AU  - Qiang Fu
AU  - Hao Song
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03272
UR  - https://arxiv.org/abs/2610.03272
AB  - 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,≥ 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≤ Mb54≤ 30 rather than the tabulated ≤ 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.
ER  -

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