Polyphase equiangular tight frames and abelian generalized quadrangles

TL;DR

This paper introduces a new method for constructing complex equiangular tight frames using abelian generalized quadrangles, linking finite geometry with signal processing applications.

Contribution

It extends previous work by constructing ETF synthesis operators from abelian generalized quadrangles, providing a new infinite family of ETFs and a novel proof of certain generalized quadrangles' existence.

Findings

Constructed new infinite family of complex ETFs

Established a link between generalized quadrangles and ETFs

Provided a new proof for the existence of specific generalized quadrangles

Abstract

An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signature matrices of ETFs were constructed from abelian distance regular covers of complete graphs. We extend this work, constructing ETF synthesis operators from abelian generalized quadrangles, and vice versa. This produces a new infinite family of complex ETFs as well as a new proof of the existence of certain generalized quadrangles. This work involves designing matrices whose entries are polynomials over a finite abelian group. As such, it is related to the concept of a polyphase matrix of a finite filter bank.