Toward the Classification of Biangular Harmonic Frames

TL;DR

This paper investigates the combinatorial structures underlying harmonic biangular tight frames, revealing conditions under which they are equivalent to difference sets or bidifference sets, and providing examples of exceptions.

Contribution

It characterizes when harmonic BTFs arise from generalized difference sets and explores their relationship with bidifference sets, including new examples and exceptions.

Findings

Harmonic BTFs generated by certain difference set generalizations are either BTFs or ETFs.

The relationship between harmonic BTFs and bidifference sets is complex and not one-to-one.

Examples of harmonic BTFs not generated by bidifference sets are provided.

Abstract

Equiangular tight frames (ETFs) and biangular tight frames (BTFs) - sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively - are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames - vector sets defined in terms of the characters of finite abelian groups - because they are characterized by combinatorial objects called difference sets. This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties - all three of which are generalizations of difference sets that fall under the umbrella term "bidifference set" - then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.