Certifying the novelty of equichordal tight fusion frames

Abstract

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.

Harmonic Grassmannian codes

“…We now give some facts that follow easily by combining Theorems 3.4, 3. It was recently shown that taking iterated alternating Naimark and spatial complements of any TFF(D, N, R) with either N > 4 or N = 4 and D = 2R yields an infinite number of TFFs with distinct parameters [17]. From Corollary 3.7, we see that if one TFF in such a Naimarkspatial orbit is harmonic then all TFFs in this orbit are as well.…”

“…) for all n 1 = n 2 , and so this TFF is real and/or equichordal if and only if its spatial complement is as well; see [17] for a more thorough discussion of these well-known phenomena.…”

“…For example, taking iterated alternating Naimark and spatial complements of the harmonic EITFF(4, 5, 2) discussed in Example 3.2 yields harmonic ECTFFs with (D, N, R) parameters (4, 5, 2), (6, 5, 2), (6,5,4), (14,5,4), (14,5,10), etc., but of these, only those with parameters (4, 5, 2) and (6, 5, 2) are EITFFs. These ideas from [17] were in part motivated by a desire to certify when a certain generalization of an ECTFF construction method of [25] yielded genuinely new ECTFFs. As we now discuss, it turns out that both this construction method [25] and its generalization [17] are essentially special cases of harmonic matrix ensembles.…”

“…These ideas from [17] were in part motivated by a desire to certify when a certain generalization of an ECTFF construction method of [25] yielded genuinely new ECTFFs. As we now discuss, it turns out that both this construction method [25] and its generalization [17] are essentially special cases of harmonic matrix ensembles.…”

“…To date, remarkably few constructions of EITFF(D, N, R) with gcd(D, R) = 1 and R > 1 are known: a real EITFF(N, N, 2) exists whenever a complex symmetric conference matrix of size N does, including whenever N is a prime power with N ≡ 1 mod 4 [11,2], and a real EITFF (8,6,3) exists [14,24]. In fact, most known EITFF(D, N, R) are "tensor-sized," having parameters that match those of one obtained by directly summing R copies of an N -vector ETF for a space of dimension D R [17]. In this paper, we generalize the celebrated equivalence [26,31,37,9] between harmonic ETFs and difference sets for finite abelian groups so as to produce an infinite number of new EITFFs, including an infinite number of new EITFF(N, N, 2) with odd N , and an EITFF (11,11,3).…”