Equiangular tight frames from hyperovals

Matthew Fickus; Dustin G. Mixon; John Jasper

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arXiv · 1602.05557

Equiangular tight frames from hyperovals

Matthew Fickus·Dustin G. Mixon·John Jasper

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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.

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Matthew Fickus, Dustin G. Mixon, John Jasper. 2016-02-17. Equiangular tight frames from hyperovals. https://arxiv.org/abs/1602.05557

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