Existence of the q=5 multispread with parameters (104,6)

Determine whether a projective 3-(152,5,{2})_5 system, equivalently a (104,6)_5^{3,5}-multispread, exists.

Background

For q=5, the paper explains that constructing a projective 3-(26+126,5,{2})_5 system would complete the characterization of the relevant additive 53-ary one-weight codes. The authors report that their searches did not find such a system, while also conjecturing a restriction on its automorphism group if it exists. The existence question therefore remains unresolved.

References

However, we have not succeeded in finding a projective $3$-$(26{+}126,5,{2})_5$ system.

— Generalized ovals, 2.5-dimensional additive codes, and multispreads  (2511.15843 - Krotov et al., 19 Nov 2025) in Section 4, “Multispreads in PG(4,q),” subsection “q=5”