Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Structure of Frames and Equiangular Lines over Finite Fields and their Connections to Design Theory

Ian Jorquera, Emily J. King

TL;DR

This work extends frame theory and equiangular line packings to finite fields by formulating nondegenerate Hermitian forms, unitary/orthogonal geometries, and a robust notion of ETFs over $ ext{F}_{q}$ and $ ext{F}_{q^{2}}$. It identifies a crucial augmentation to Welch-bound saturation: beyond bound equality, sums of triple products must meet precise criteria to guarantee ETF-ness, and it shows how regular simplices and Naimark complements encode ETF structure via discriminants. The paper further develops switching equivalence in finite-field settings, connects ETFs to two-graphs and quasi-symmetric designs, and threads these ideas through to real-field realizations and incoherence phenomena. The resulting framework yields explicit conditions for tightness, designs arising from ETF substructures, and a rich interplay between algebraic and combinatorial objects in orthogonal geometries, with implications for coding, design theory, and finite-field signal processing.

Abstract

This paper concerns frames and equiangular lines over finite fields. We find a necessary and sufficient condition for systems of equiangular lines over finite fields to be equiangular tight frames (ETFs). As is the case over subfields of $\mathbb{C}$, it is necessary for the Welch bound to be saturated, but there is an additional condition required involving sums of triple products. We also prove that similar to the case over $\mathbb{C}$, collections of vectors are similar to a regular simplex essentially when the triple products of their scalar products satisfy a certain property. Finally, we investigate switching equivalence classes of frames and systems of lines focusing on systems of equiangular lines in finite orthogonal geometries with maximal incoherent sets, drawing connections to combinatorial design theory.

On the Structure of Frames and Equiangular Lines over Finite Fields and their Connections to Design Theory

TL;DR

This work extends frame theory and equiangular line packings to finite fields by formulating nondegenerate Hermitian forms, unitary/orthogonal geometries, and a robust notion of ETFs over and . It identifies a crucial augmentation to Welch-bound saturation: beyond bound equality, sums of triple products must meet precise criteria to guarantee ETF-ness, and it shows how regular simplices and Naimark complements encode ETF structure via discriminants. The paper further develops switching equivalence in finite-field settings, connects ETFs to two-graphs and quasi-symmetric designs, and threads these ideas through to real-field realizations and incoherence phenomena. The resulting framework yields explicit conditions for tightness, designs arising from ETF substructures, and a rich interplay between algebraic and combinatorial objects in orthogonal geometries, with implications for coding, design theory, and finite-field signal processing.

Abstract

This paper concerns frames and equiangular lines over finite fields. We find a necessary and sufficient condition for systems of equiangular lines over finite fields to be equiangular tight frames (ETFs). As is the case over subfields of , it is necessary for the Welch bound to be saturated, but there is an additional condition required involving sums of triple products. We also prove that similar to the case over , collections of vectors are similar to a regular simplex essentially when the triple products of their scalar products satisfy a certain property. Finally, we investigate switching equivalence classes of frames and systems of lines focusing on systems of equiangular lines in finite orthogonal geometries with maximal incoherent sets, drawing connections to combinatorial design theory.
Paper Structure (17 sections, 40 theorems, 81 equations)

This paper contains 17 sections, 40 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Frames over Arbitrary Fields and Combinatorial Designs
  3. Foundations Remixed
  4. Frames
  5. Equiangular Lines and ETFs
  6. Switching Equivalence
  7. Combinatorial Designs: Two-Graphs and Quasi-Symmetric Designs
  8. Sufficient Conditions for Tightness
  9. Lemma 2.4 Revisited
  10. Naimark Complements
  11. Tightening Equiangularity
  12. Two-graphs and ETFs in Orthogonal Geometries
  13. Simplices: Equiangular Cliques
  14. ETFs with Respectable Characteristics
  15. ETFs with Defiant Characteristics
  16. ...and 2 more sections

Key Result

Lemma 2.2

If $\left\langle -,-\right\rangle$ is a non-degenerate Hermitian scalar product for $V$, then there exists a basis $v_1,\dots,v_d$ where $\left\langle v_j,v_j\right\rangle=b_j$, for $b_j\in\mathbb{F}_0^\times$ and all other products between basis elements are zero. Furthermore, in Case U, it can be

Theorems & Definitions (93)

  • Remark 2.1
  • Lemma 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 83 more