论文标题
有限字段的帧：基本理论和统一线中的曲线
Frames over finite fields: Basic theory and equiangular lines in unitary geometry
论文作者
论文摘要
我们介绍了有限场上经典几何形状中的框架和等法线的研究。在制定基本理论之后，我们提供了几个示例，并证明了由模块化差异集以及翻译和调制算子产生的等效紧密帧（ETF）的有限场类似物。使用后者，我们证明了Gerzon的界限是在dimension $ d = 2^{2l+1} $上的每个单位几何形状中获得的，字段$ \ mathbb {f} _ {3^2} $。我们还研究了复杂的ETF与有限的单一几何形状的相互作用，我们表明，每个复杂的ETF都意味着在无限许多有限领域的ETF存在相同大小的ETF。
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.
