Equiangular tight frames from group divisible designs

Abstract

An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs. The real instances of these ETFs correspond to several new infinite families of strongly regular graphs. Our approach was inspired by a seminal paper of Davis and Jedwab which both unified and generalized McFarland and Spence difference sets. We provide combinatorial analogs of their algebraic results, unifying Steiner ETFs with hyperoval ETFs and Tremain ETFs.

Certifying the novelty of equichordal tight fusion frames

“…3 ) yields real TFF(D, 3, R) with R = D 2 and R = 2D 3 , respectively. From this result, we see that only ever at most a small portion of an orbit of the form (16) or (17) will correspond to actual TFFs. In particular, the parameters of any TFF(D, 2, R) appear in one of two types of "Naimark-spatial paths" of length two:…”

“…Their (D ′ , R ′ ) parameters are obtained by moving horizontally or vertically, respectively, from (D, R) to another point on this hyperbola. Interestingly, the path from (3, 1) to (4, 1) to (4, 3) to (17,3), etc., (alternating complements, beginning with Naimark) "interweaves" with that from (3, 1) to (3,2) to (11,2) to (11,9), etc., (alternating complements, beginning with spatial). From this graph, it is clear that Orb(3, 7, 1) is infinite and moreover contains a unique point (D 0 , 7, R 0 ) such that D 0 ≤ D and R 0 ≤ R for all (D, 7, R) ∈ Orb(3, 7, 1), namely (D 0 , 7, R 0 ) = (3, 7, 1).…”

“…In either case, a trivial real TFF(D, N, R) exists. Similarly if N = 3 and R 0 = D 0 3 then (17) gives that Orb(3R 0 , 3, R 0 ) is {(3R 0 , 3, R 0 ), (0, 3, R 0 ), (0, 3, −R 0 ), (−3R 0 , 3, −R 0 ), (−3R 0 , 3, −2R 0 ), (3R 0 , 3, 2R 0 )}, implying (D, N, R) is either (3R 0 , 3, R 0 ) or (3R 0 , 3, 2R 0 ), and so by Theorem 3.2 that a real TFF(D, N, R) exists. When N = 3 and R 0 = D 0 , (17) instead gives that Orb(R 0 , 3, R 0 ) is n=1 is a TFF(2R, 4, R) for R 2R .…”

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].…”

Harmonic equiangular tight frames comprised of regular simplices

“…Meanwhile, Steiner ETFs arise from balanced incomplete block designs [26,25]. This construction has recently been generalized to yield new infinite families of ETFs arising from projective planes that contain hyperovals, Steiner triple systems, and group divisible designs [24,21,19].…”