Parallelisms of \\mathrmPG(3, 4) Invariant Under Cyclic Groups of Order 4

Abstract

A spread in \\(\\mathrm{PG}(n,q)\\) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, cryptography, etc. All parallelisms of \\(\\mathrm{PG}(3,2)\\) and \\(\\mathrm{PG}(3,3)\\) are known and parallelisms of \\(\\mathrm{PG}(3,4)\\) which are invariant under automorphisms of odd prime orders and under the Baer involution have already been classified. In the present paper, we classify all (we establish that their number is 252738) parallelisms in \\(\\mathrm{PG}(3,4)\\) that are invariant under cyclic automorphism groups of order 4. We compute the order of their automorphism groups and obtain invariants based on the type of their spreads and duality.