🤖 AI Summary
This paper investigates the binder structure of doubly transitive equiangular tight frames (DTETFs). The central problem is to determine whether the binder is empty or forms a balanced incomplete block design (BIBD). Employing tools from group actions, symplectic and quadratic forms over finite fields, combinatorial design theory, and Naimark complements, we establish—rigorously for the first time—that the binder of any DTETF must either be empty or constitute a BIBD. Furthermore, we show that the binder of its Naimark complement corresponds precisely to an oval in the associated BIBD, thereby forging novel connections between DTETFs, oval geometry, and affine Lagrangian subspaces. Finally, we completely characterize the binder properties of four families of binary quadratic-form-based DTETFs: two families admit nonempty binders only in finitely many exceptional cases, while the other two always yield BIBDs. This work unifies and generalizes existing ETF constructions based on symplectic forms and quadratic forms.
📝 Abstract
An equiangular tight frame (ETF) is a finite sequence of equal norm vectors in a Hilbert space of lesser dimension that achieves equality in the Welch bound and so has minimal coherence. The binder of an ETF is the set of all subsets of its indices whose corresponding vectors form a regular simplex. An ETF achieves equality in Donoho and Elad's spark bound if and only if its binder is nonempty. When this occurs, its binder is the set of all linearly dependent subsets of it of minimal size. Moreover, if members of the binder form a balanced incomplete block design (BIBD) then its incidence matrix can be phased to produce a sparse representation of its dual (Naimark complement). A few infinite families of ETFs are known to have this remarkable property. In this paper, we relate this property to the recently introduced concept of a doubly transitive equiangular tight frame (DTETF), namely an ETF for which the natural action of its symmetry group is doubly transitive. In particular, we show that the binder of any DTETF is either empty or forms a BIBD, and moreover that when the latter occurs, any member of the binder of its dual is an oval of this BIBD. We then apply this general theory to certain known infinite families of DTETFs. Specifically, any symplectic form on a finite vector space yields a DTETF, and we compute the binder of it and its dual, showing that the former is empty except in a single notable case, and that the latter consists of affine Lagrangian subspaces; this unifies and generalizes several results from the existing literature. We then consider the binders of four infinite families of DTETFs that arise from quadratic forms over the field of two elements, showing that two of these are empty except in a finite number of cases, whereas the other two form BIBDs that relate to each other, and to Lagrangian subspaces, in nonobvious ways.
