Equi-isoclinic subspaces from difference sets

Equi-chordal tight fusion frames (ECTFFs) and equi-isoclinic tight fusion frames (EITFFs) are types of optimal packings in Grassmannian spaces. In particular, an ECTFF is an arrangement of equi-dimensional subspaces of a Euclidean space with the property that the smallest chordal distance between any pair of these subspaces is as large as possible. An EITFF is a special type of ECTFF that also happens to be an optimal packing with respect to the spectral distance. In the special case where the subspaces are one-dimensional, both ECTFFs and EITFFs reduce to optimal packings of lines known as equiangular tight frames (ETFs); these lines have minimal coherence, achieving equality in the Welch bound. ETFs are tricky to construct, but several infinite families of them are known. Harmonic ETFs in particular arise by restricting the characters of a finite abelian group to a difference set for that group. Moreover, there is a simple tensor-based method for combining ETFs with orthonormal bases to form EITFFs. It is an open question as to whether every EITFF essentially arises in this way.In this short paper, we preview a new result relating difference sets, harmonic ETFs, ECTFFs and EITFFs. This work expands on other recent results showing that certain harmonic ETFs are comprised of a number of simpler ETFs known as regular simplices; such ETFs arise, for example, from McFarland difference sets as well as the complements of certain Singer difference sets. It is already known that in this situation the subspaces spanned by these regular simplices necessarily form an ECTFF. We recently discovered that these same subspaces form an EITFF in some, but not all, cases. In an upcoming journal article, we shall characterize the properties of the underlying difference sets that lead to EITFFs in this fashion.