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.

我们介绍了有限域上经典几何中的框架和等角线的研究。在发展了基本理论之后，我们给出了几个例子并演示了由模块化差分集以及平移和调制算子产生的等角紧框架 (ETF) 的有限域模拟。使用后者，我们证明了 Gerzon 的界限在每个单一的维度几何中都可以达到$d=2^{2升+1}$ 在场上 ${\mathbb{F}}_{3^2}$. 我们还研究了复杂 ETF 与有限酉几何中的 ETF 之间的相互作用，并且我们表明，每个复杂 ETF 都意味着在无限多个有限域中存在相同大小的 ETF。