Existence of the q=3 strongly regular graph

Determine whether a strongly regular graph with parameters (3^5, (3^2-1)(3^2+3+2), 3^3+2\cdot3^2+3-2, 3^3+2\cdot3^2+3\cdot3+2) exists, equivalently whether the corresponding parameter case discussed for q=3 can be realized.

Background

The paper derives a strongly regular graph parameter set from a hypothetical faithful projective 2-(q2+q+2,5,2)_q system when q is odd. For q=3, the resulting graph parameters are an unresolved case in an external catalogue. The authors note that related examples do exist for q=4, but do not resolve the q=3 case.

References

Interestingly enough, the case for $q=3$ is marked as open at \url{https://aeb.win.tue.nl/graphs/srg/srgtab201-250.html}.

— Generalized ovals, 2.5-dimensional additive codes, and multispreads  (2511.15843 - Krotov et al., 19 Nov 2025) in Section 2, subsection “At most two subspaces in a hyperplane — generalized ovals”