Parallelisms of  \(\textrm{PG}(3,4)\) with a Great Number of Regular Spreads

A spread in the finite projective space \(\textrm{PG}(n,q)\) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. Parallelisms with many regular spreads are of particular interest. All parallelisms of \(\textrm{PG}(3,4)\) which are invariant under automorphisms of orders greater than 2 and some of the parallelisms with automorphisms of order 2 are known. Among them there are no parallelisms with more than 13 regular spreads (out of all 21 spreads). To establish whether a parallelism of \(\textrm{PG}(3,4)\) with more than 13 regular spreads exists was an open problem before the present work. We construct all parallelisms with automorphisms of order 2 and at least 13 regular spreads and succeed to improve the previous result by finding out that there exist parallelisms of \(\textrm{PG}(3,4)\) with 16 regular spreads.