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We introduce the study of frames and equiangular lines in classicalgeometries over finite fields. After developing the basic theory, we giveseveral examples and demonstrate finite field analogs of equiangular tightframes (ETFs) produced by modular difference sets, and by translation andmodulation operators. Using the latter, we prove that Gerzon's bound isattained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field$\mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs andthose in finite unitary geometries, and we show that every complex ETF impliesthe existence of ETFs with the same size over infinitely many finite fields.

Frames over finite fields: Basic theory and equiangular lines in unitary geometry