复空间ℂᵈ中2d条直线的最优排列的新构造
New constructions of optimal arrangements of $2d$ lines in $\mathbb{C}^d$
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中文总结 AI 辅助
该研究将Fallon等人的加倍构造推广为张量乘法构造，结合阿达马矩阵与等角紧框架配对，基于素数幂等给出复空间中等角紧框架的新构造，推广了Turyn的会议矩阵构造。
中文摘要 AI 辅助
本文中，我们给出了复空间ℂᵈ中大小为2d的等角紧框架的新构造。我们将Fallon和Iverson的加倍构造推广为基于合适配对的张量乘法构造，该配对由复阿达马矩阵与等角紧框架组成。特别地，只要存在实阿达马矩阵的友好配对，此类配对就始终存在。最值得注意的是，对于所有满足q≡3（mod 4）的素数幂q，存在阶为q+1的友好阿达马配对。我们还基于阶为6的配对族，以及等角紧框架由满足q≡1（mod 4）的Paley会议矩阵定义的配对，找到了具体构造。最后，我们给出了等角紧框架的幂构造，该构造推广了Turyn的会议矩阵构造。
英文摘要
In this paper we provide new constructions of equiangular tight frames of size $2d$ in $\mathbb{C}^d$. We generalize the doubling construction of Fallon and Iverson to a tensor multiplication construction based on a suitable pair consisting of a complex Hadamard matrix and an equiangular tight frame. In particular, such a pair always exists whenever there is an amicable pair of real Hadamard matrices. Most notably, amicable Hadamard pairs of order $q+1$ exist for all prime powers $q\equiv 3\pmod 4$. We also find specific constructions based on a family of pairs of order 6 and on pairs whose equiangular tight frames are defined by Paley conference matrices with $q\equiv 1\pmod 4$. Finally, we provide a power construction of equiangular tight frames that generalizes the construction of Turyn for conference matrices.
