Maximum dimension for hemisystems in Hermitian polar spaces

Determine the maximal integer k such that hemisystems with respect to k-spaces exist in the Hermitian polar space (2d−1,q²) for d≥3.

Background

The paper establishes that, for d>2 and q≠3, the Hermitian polar space (2d−1,q²) has no hemisystems with respect to (d−2)-spaces. It also notes that hemisystems with respect to points do exist. The unresolved problem is therefore to identify the largest dimension k for which such hemisystems can exist when d≥3, thereby determining the precise upper limit on the dimensions supporting hemisystems.