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.
Highlights
A partial (t − 1)-spread of a projective or polar space P is a set S of pairwise disjoint (t − 1)-dimensional subspaces of P
In this paper we investigate partial spreads of H(2n − 1, q2) 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, q2)
Summary
A partial (t − 1)-spread of a projective or polar space P is a set S of pairwise disjoint (t − 1)-dimensional subspaces of P. A partial spread set U is a set of n × n matrices over a field F such that (i) rank(A − B) = n for all A, B ∈ U , A = B;. A constant rank-distance n set is a partial spread set. Partial spread set) is said to be maximal if it is not strictly contained in a larger constant rank-distance k set Partial spread set) is closed under F -linear combinations for some field F , we refer to it as an F -linear constant rank-distance k set
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