A computer search of maximal partial spreads in PG(3,q)

TL;DR

This paper uses computational methods to determine minimal sizes of maximal partial spreads in projective spaces PG(3,q) for various q, providing new bounds and complete classifications for several cases.

Contribution

It presents the first computational search results for minimal sizes of maximal partial spreads in PG(3,q) for multiple q values, including new bounds and classifications.

Findings

New minimum sizes for q=8,9,16 and q between 25 and 101.

Complete classification of sizes for q=8,9,16,25,27.

Confirmed the size 45 for q=7.

Abstract

In this work we find new minimum sizes for the maximal partial spreads of PG$(3,q)$, for $q=8,9,16$ and for every $q$ such that $25\leq q\leq 101$. Furthermore, for $q=8,9,16,25$ and 27 we find all the unknown sizes between our minimums and the value $q^{2}-q+2$. Moreover, we obtain density results also in the cases $q=19$ and $q=23$, already studied but not yet completed. Finally, we find the known exceptional size 45 for $q=7$.