Codes and Expansions (CodEx) Seminar

Kean Fallon (Iowa State University):
Symplectic Zauner's Conjecture is the Skew Hadamard Conjecture

Consider the problem of placing lines through the origin so that each of the pairwise angles between them is the same. When in a complex vector space, Gerzon's bound states the largest number of these equiangular lines one can have is the square of the dimension of the space you are in. Zauner's conjecture, a significant open problem that has garnered considerable interest, speculates that this bound is saturated in every dimension.

We introduce an analogous notion of equiangular lines and, subsequently, equiangular tight frames, over real symplectic space. In particular, when porting Zauner's conjecture to real symplectic setting, we find that the symplectic Zauner's conjecture is equivalent to the skew Hadamard conjecture. In this talk, we detail this equivalence, the proof of which relies on identifying and leveraging a class of graphs underlying every equiangular tight frame in real symplectic space.