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.

翻译：等角紧框架（ETF）是希尔伯特空间中一组等范数向量构成的有限序列，其维数小于空间维数，且达到韦尔奇下界，因而具有最小相干性。ETF的绑定集是指标子集的集合，使得对应向量构成正则单形。ETF达到多诺霍-埃拉德稀疏界当且仅当其绑定集非空。当此条件成立时，绑定集即为所有最小规模的线性相关子集。此外，若绑定集成员构成平衡不完全区组设计（BIBD），则其关联矩阵可通过相位调整得到其对偶（奈马克补）的稀疏表示。已知少数无限族ETF具有这一显著性质。本文将这一性质与最近提出的双可迁等角紧框架（DTETF）概念联系起来——DTETF是指其对称群的自然作用为双可迁的ETF。具体而言，我们证明任意DTETF的绑定集要么为空，要么构成BIBD；且当后者成立时，其对偶绑定集中的任意成员均为该BIBD的卵形。随后我们将这一通用理论应用于若干已知无限族DTETF。特别地，有限向量空间上的任意辛形式均可生成DTETF，我们计算了该DTETF及其对偶的绑定集，证明前者仅在一种特殊情形下非空，而后者由仿射拉格朗日子空间构成；这统一并推广了现有文献中的若干结论。接着我们考虑由二元域上二次型导出的四类无限族DTETF的绑定集，证明其中两类仅在有限个情形下非空，而另两类构成BIBD，且彼此间以及与拉格朗日子空间之间存在非平凡关联。