Frames over finite fields: Basic theory and equiangular lines in unitary geometry

TL;DR

This paper develops the theory of frames and equiangular lines over finite fields, providing examples, finite field analogs of ETFs, and demonstrating the attainment of Gerzon's bound in certain finite unitary geometries.

Contribution

It introduces the basic theory of frames over finite fields, constructs finite field analogs of ETFs, and proves Gerzon's bound is attained in specific finite unitary geometries.

Findings

Finite field analogs of equiangular tight frames are constructed.

Gerzon's bound is achieved in certain finite unitary geometries.

Every complex ETF implies the existence of ETFs over infinitely many finite fields.

Abstract

We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field $\mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.