Paper detail

Certifying the novelty of equichordal tight fusion frames

An equichordal tight fusion frame (ECTFF) is a finite sequence of equi-dimensional subspaces of a finite-dimensional Hilbert space that achieves equality in Conway, Hardin and Sloane's simplex bound. Every ECTFF 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 ECTFF has both a Naimark complement and spatial complement which themselves are ECTFFs. It turns out that whenever the number of subspaces is at least five, taking iterated alternating Naimark and spatial complements of one ECTFF yields an infinite family of them with distinct parameters. This makes it challenging to certify the novelty of any recently discovered ECTFF: how can one guarantee that it does not arise from any previously known construction in such a Naimark-spatial way? In this paper, we propose a solution to this problem, showing that any ECTFF is a member of a Naimark-spatial family originating from either a trivial ECTFF or one with unique "minimal" parameters. In the latter case, if its minimal parameters do not match those of any previously known ECTFF, it is certifiably new. As a proof of concept, we then use these ideas to certify the novelty of some ECTFFs arising from a new method for constructing them from difference families for finite abelian groups. This method properly generalizes King's construction of ECTFFs from semiregular divisible difference sets.