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

Source 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.