Classification of collineation orbits of complete 21-point sets

Classify the collineation orbits of complete 21-point 4-general sets in PG(5,4), in order to test the intersection bound against one representative from each orbit.

Background

The paper proves an intersection bound for a hypothetical 22-point 4-general set relative to the known complete 21-point configuration and, by orbit invariance, relative to every configuration in its PGL(6,4)-orbit. Only one collineation orbit of complete 21-point 4-general sets is known to the authors.

The proposed strengthening is to classify all such orbits and apply the rigidity argument to each representative. This could support an unconditional proof that no 22-point set exists, but the classification itself remains unresolved.

References

The aim is the unconditional assertion that no $22$-point set exists. Two further questions are to classify those orbits and to decide whether the constructions of can be beaten for $d\ge3$.

— 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'