Greaves et al. (2022) extended frames over real or complex numbers to frames over finite fields. In this paper, we study the theory of frames over finite fields by incorporating the Galois inner products introduced by Fan and Zhang (2017), which generalize the Euclidean and Hermitian inner products. We define a class of frames, called Galois frames over finite fields, along with related notions such as Galois Gram matrices, Galois frame operators, and Galois equiangular tight frames (Galois ETFs). We also characterize when Galois self-dual codes induce Galois ETFs. Furthermore, we construct explicitly Galois ETFs from Galois self-dual constacyclic codes.

翻译：Greaves等人（2022）将实数域或复数域上的框架推广到了有限域上的框架。本文通过引入Fan和Zhang（2017）提出的Galois内积（该内积推广了欧几里得内积和厄米特内积），研究了有限域上框架的理论。我们定义了一类称为有限域上Galois框架的结构，以及相关的概念，如Galois Gram矩阵、Galois框架算子和Galois等角紧框架（Galois ETFs）。我们还刻画了Galois自对偶码何时能诱导出Galois ETFs。此外，我们利用Galois自对偶常循环码显式地构造了Galois ETFs。