The climb problem over F5

Determine whether the Pavese constructions of 4-general sets over F5 can be improved by one point; equivalently, resolve the corresponding unresolved climb problem over F5.

Background

The paper determines the extremal value 34_5=12 for PG(4,5), but explains that the near-MDS reduction used over F3 and F4 does not apply over F5. In particular, the extremal 12-point sets are not NMDS-sets, so the near-MDS bridge cannot settle the associated climb questions.

The authors explicitly state that the climb question remains open over F5, leaving unresolved whether the relevant constructions can be improved at the dimensions and parameters not settled by the code tables.

References

The bridge of Lemma~\ref{lem:large4gen} fails over $F_5$ (Remark~\ref{rem:nogens}) because $[6,2,\ge5]_5$ exists, so the cell cannot be read off from near-MDS codes at all, and the climb question remains open over $F_5$ as well.

— The climb problem for $4$-general sets in PG(n,4)  (2610.03272 - Lu et al., 2 Oct 2026) in Section 5, 'Concluding remarks and open problems'