  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.
