Equiangular Tight Frames From Hyperovals
Equiangular Tight Frames From Hyperovals
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DOI：
10.1109/tit.2016.2587865
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发表时间：
2016-02
影响因子：
2.5
通讯作者：
M. Fickus;D. Mixon;J. Jasper
中科院分区：
文献类型：
--
作者：
M. Fickus;D. Mixon;J. Jasper
等角紧框架（ETF）是欧几里得空间中相干性尽可能小的相等范数向量的集合，等于韦尔奇界。这种帧也被称为韦尔奇界相等序列，出现在各种应用中，如波形设计、量子信息理论、压缩感知和代数编码理论。etf似乎很少见，目前已知的构建etf的方法也不多。本文给出了由有限投影平面上的超卵圆产生的一类新的无限复etf族。特别地，我们给出了19维空间中76个向量的复ETF的第一个构造。最近，一种计算机辅助方法被用来证明这种规模的真正etf并不存在，解决了该领域长期存在的开放性问题。我们的构建是对先前已知的从平衡不完全块设计构建etf的技术的修改。
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.