Galois equiangular tight frames from Galois self-dual codes

TL;DR

This paper develops the theory of Galois equiangular tight frames over finite fields using Galois inner products, characterizes their construction from self-dual codes, and provides explicit examples.

Contribution

It introduces Galois frames over finite fields, characterizes when Galois self-dual codes induce Galois ETFs, and constructs explicit Galois ETFs from Galois self-dual constacyclic codes.

Findings

Galois frames generalize classical frames over finite fields.

Galois self-dual codes can induce Galois ETFs.

Explicit constructions of Galois ETFs from specific codes.

Abstract

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.