Existence of a 22-point 4-general set in PG(5,4)

Determine whether the projective space PG(5,4) contains a 22-point 4-general set, equivalently whether a quaternary [22,16,5] code exists.

Background

The paper establishes a 21-point 4-general construction in PG(5,4), giving the lower bound 21 ≤ 35_4, but the code tables do not exclude a 22-point set. By the reduction developed earlier, a 22-point 4-general set in PG(5,4) is equivalent to a projective quaternary [22,16,5] code. The authors report the verified range 21 ≤ 35_4 ≤ 30 and note that no known bound excludes 22.

A rigidity computation shows that any hypothetical 22-point set intersects the published extremal 21-point configuration in at most 12 points, but this does not resolve existence. Establishing nonexistence would prove that the 21-point construction is optimal at this rung.

References

The open question of the ladder is whether $PG(5,4)$ contains a $22$-point $4$-general set, equivalently whether a $[22,16,5]_4$ code exists, equivalently whether $\mathrm{Lb}(22,16)=4$ improves to $5$.

— 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'

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'