Cardinality of the smallest larger-than-projected-subgeometry blocking set

Determine the value of δ′, defined by requiring p^3+δ′ to be 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

For q=p3, the paper defines a threshold δ_0 using δ′, where δ′ measures the size excess over p3 of the smallest non-trivial minimal blocking set in PG(2,p3) larger than the projected-subgeometry size p3+p2+p+1.

The exact value of δ′ is needed to specify the strongest available deficiency bounds for maximal partial spreads in PG(3,p3), and consequently for the corresponding classification results for Cameron–Liebler sets on Q+(5,p3). The paper gives an upper bound but states that the value itself is unresolved.

References

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

— Cameron-Liebler sets of generators in the Klein quadric $Q^+(5,q)$  (2503.08260 - D'haeseleer et al., 11 Mar 2025) in Section 7, notation preceding Theorem 7.1 (the definition of δ_0 for q=p^3)