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). 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,q2) for a
range of sizes