New Examples of Maximal Partial Line Spreads in \(PG(4, q)\)

Sandro Rajola1, Maurizio Iurlo2
1Istituto Tecnico per il Turismo “C. Colombo” Via Panisperna, 255 00184 Roma Italy
2Largo dell’ Olgiata, 15/106/1C 00123 Roma Italy

Abstract

We construct a class of maximal partial line spreads in \( \mathrm{PG}(4, q) \), that we call \( q \)-added maximal partial line spreads. We obtain them by depriving a line spread of a hyperplane of some lines and adding \( q+1 \) pairwise skew lines not in the hyperplane for each removed line. We do it in a theoretical way for every value of \( q \), and by a computer search for \( q \leq 16 \). More precisely, we prove that for every \( q \) there are \( q \)-added MPS of size \( q^2 + kq + 1 \), for every integer \( k = 1, \ldots, q-1 \), while by a computer search we get larger cardinalities.

Keywords: Maximal partial line spreads