Equiangular tight frames from group divisible designs

TL;DR

This paper introduces a new method to construct larger equiangular tight frames (ETFs) using group divisible designs, leading to several new infinite families of ETFs and related strongly regular graphs.

Contribution

It presents a novel technique combining existing ETFs with group divisible designs to generate larger ETFs, expanding known families and unifying various ETF constructions.

Findings

Several new infinite families of ETFs were constructed.

New strongly regular graphs corresponding to real ETFs were identified.

The method unifies multiple existing ETF constructions through combinatorial analogs.

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.