Smallest non-trivial minimal blocking set beyond the known projected-subgeometry size

Determine the exact value of δ′, defined by the property that p^3+δ′ is the cardinality of the smallest non-trivial minimal blocking set in PG(2,p^3) whose cardinality exceeds p^3+p^2+p+1.

Background

The paper imports results on minimal blocking sets in projective planes to obtain structural results for holes of maximal partial spreads in PG(3,p3), and consequently for Cameron–Liebler sets on Q+(5,p3). The threshold δ′ is used in defining the parameter δ_0 governing those structural results.

The known projected-subgeometry constructions produce minimal blocking sets of size p3+p2+p+1. The next possible size of a non-trivial minimal blocking set is not determined in the stated setting. Resolving δ′ would sharpen the value of δ_0 and therefore potentially extend the range in which holes, and the corresponding Cameron–Liebler sets, can be classified.

References

Presently, this value is still unknown, but we know δ' \leq p3/p_0+1.

— Cameron-Liebler sets of generators in the Klein quadric $Q^+(5,q)$  (2503.08260 - D'haeseleer et al., 11 Mar 2025) in Section 5, notation preceding Theorem 5.5 (Theorems labeled final1 and final2)