Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces

TL;DR

This paper studies the structure and limitations of certain matrix sets and partial spreads in hermitian polar spaces, establishing bounds and constructing examples to advance understanding in finite geometry.

Contribution

It provides a tight upper bound on linear constant rank-distance sets of hermitian matrices and demonstrates the maximality of certain partial spreads in hermitian polar spaces.

Findings

Established a tight upper bound for linear constant rank-distance sets.

Proved the maximality of extensions of symplectic semifield spreads.

Constructed large examples and maximal partial spreads for specific cases.

Abstract

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes.