Galois equiangular tight frames from Galois self-dual codes

2025-07-21

AI Summary

This work addresses the algebraic construction of equiangular tight frames (ETFs) over finite fields. To this end, it introduces the Galois inner product to establish a novel framework theory—defining Galois frames, Galois Gram matrices, and Galois frame operators. The core method establishes a necessary and sufficient condition under which Galois self-dual codes induce Galois ETFs, and explicitly constructs multiple families of Galois ETFs from Galois self-dual quasi-cyclic codes for the first time. These results unify and generalize classical Euclidean and Hermitian ETF constructions, deeply integrating algebraic coding theory—particularly self-duality—into finite-field frame design. The proposed framework provides new algebraic tools and systematic construction methods for structured measurement matrices in low-dimensional embeddings, compressed sensing, and quantum information theory.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Game Theory and Economic Paradigms: Mechanism Design
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Extend frames to finite fields using Galois inner products
• Define Galois frames and related operators over finite fields
• Construct Galois ETFs from Galois self-dual constacyclic codes

Innovation

• Extend frames to finite fields
• Introduce Galois inner products
• Construct Galois ETFs explicitly