Harmonic equiangular tight frames comprised of regular simplices

Abstract

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies.

Harmonic Grassmannian codes

“…Dihedral complex symmetric conference matrices are also known to exist for any M ∈ {2, 5, 6, 8, 9, 10, 14, 17, 18}, but of these, only the M = 8 case is particularly notable since (real) symmetric conference matrices of these sizes are known to exist whenever M ∈ {2, 6, 10, 14, 18}, and complex symmetric conference matrices of these sizes arises from (real) symmetric conference matrices of size M + 1 when M ∈ {5, 9, 17}. We also note that [19] provides a circulant complex conference matrix of size M whenever M − 1 is a prime power, but these are not necessarily symmetric or skew-symmetric.…”

“…Taking (19) as our definition of π, note that for any γ 1 , γ 2 ∈ G, our block-circulant assumption gives that…”

“…In [17] it was more generally shown that {Φ κ } κG ⊥ ∈ K/G ⊥ are the isometries of an ECTFF(D, G, R) for C D if and only if the subsets {D (k) } k+G∈K/G of G all have the same cardinality and form a difference family for G, having k+G∈K/G |(Γ * 1 D (k) )(γ)| 2 be constant over all γ ∈ G, γ = 1; they are moreover the isometries of an EITFF(D, G, R) for C D if and only if each D (k) is a difference set for G. (A similar idea has also been used to partition certain harmonic ETFs into regular simplices [15,19] or, more generally, into mutually unbiased ETFs [16].) Such TFFs are "harmonic" in the sense that their isometries are constructed by extracting rows from the character table of a finite abelian group.…”

“…(A.20)The right-hand equality of (A.20) involves trigonometry: since exp( 2πi 5 ) = a and exp( 2πi 10 ) = −a 3 , sin( 2π 5 ) cos( 2π 10 ) = 1 i (a − a 4 )(−a 3 − a 2 );combining these identities with the fact that 0 = 4 n=0 a n gives −2(a+ a 4 ) + (a − a 4 )(−a 3 − a 2 ) = −a + a 2 − a 3 − 3a 4 = 1 + 2a 2 − 2a 4 ,as claimed. In light of this, (A 19…”

Testing isomorphism between tuples of subspaces

“…For example, one may use strongly regular graphs to obtain optimal codes in Gr(1, R d ) [66], or use Steiner systems to obtain optimal codes in Gr(1, C d ) [28]. In this spirit, several infinite families of optimal codes have been constructed from combinatorial designs [40,44,27,24,25,23,20,29,19]. In some cases, it is even possible to construct optimal codes from smaller codes [5,63,6,43].…”