Equiangular tight frames from hyperovals

TL;DR

This paper introduces a novel infinite family of complex equiangular tight frames derived from hyperovals in finite projective planes, including the first known ETF of 76 vectors in 19 dimensions, advancing the construction methods in the field.

Contribution

It presents a new construction method for complex ETFs using hyperovals, including the first ETF of its size and dimension, expanding the known classes of ETFs.

Findings

First construction of a complex ETF with 76 vectors in 19-dimensional space.

New infinite family of complex ETFs from hyperovals.

Resolution of a longstanding open problem regarding real ETFs of this size.

Abstract

An equiangular tight frame (ETF) is a set of equal norm vectors in a Euclidean space whose coherence is as small as possible, equaling the Welch bound. Also known as Welch-bound-equality sequences, such frames arise in various applications, such as waveform design, quantum information theory, compressed sensing and algebraic coding theory. ETFs seem to be rare, and only a few methods of constructing them are known. In this paper, we present a new infinite family of complex ETFs that arises from hyperovals in finite projective planes. In particular, we give the first ever construction of a complex ETF of 76 vectors in a space of dimension 19. Recently, a computer-assisted approach was used to show that real ETFs of this size do not exist, resolving a longstanding open problem in this field. Our construction is a modification of a previously known technique for constructing ETFs from balanced incomplete block designs.