A new class of maximal partial spreads in PG(4,q)

TL;DR

This paper introduces a new class of maximal partial spreads in projective geometry, called q-added spreads, constructed theoretically for all q and computationally for specific small q, expanding the known sizes of such spreads.

Contribution

It presents a novel construction method for maximal partial spreads in PG(4,q), extending the known sizes and providing both theoretical and computational results.

Findings

Existence of q-added maximal partial spreads for all q.

Range of sizes from q^2+q+1 to q^2+(q-1)q+1.

Computational evidence for larger sizes at small q.

Abstract

In this work we construct a new class of maximal partial spreads in $PG(4,q)$, that we call $q$-added maximal partial spreads. We obtain them by depriving a spread of a hyperplane of some lines and adding $q+1$ lines not of the hyperplane for each removed line. We do this in a theoretic way for every value of $q$, and by a computer search for $q$ an odd prime and $q \leq 13$. More precisely we prove that for every $q$ there are $q$-added maximal partial spreads from the size $q^2+q+1$ to the size $q^2+(q-1)q+1$, while by a computer search we get larger cardinalities.