Naimark-spatial families of equichordal tight fusion frames

An equichordal tight fusion frame (

) is a finite sequence of equi-dimensional subspaces of a Euclidean space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every

is a type of optimal Grassmannian code, being a way to arrange a given number of members of a Grassmannian so that the minimal chordal distance between any pair of them is as large as possible. Any nontrivial

has both a Naimark complement and spatial complement which themselves are

s. We show that taking iterated alternating Naimark and spatial complements of any

of at least five subspaces yields an infinite family of

s with pairwise distinct parameters. Generalizing a method by King, we then construct

s from difference families for finite abelian groups, and use our Naimark-spatial theory to gauge their novelty.