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

TL;DR

This paper characterizes when systems of equiangular lines over finite fields form equiangular tight frames, linking algebraic conditions to design theory and extending known results from complex spaces.

Contribution

It provides necessary and sufficient conditions for equiangular lines over finite fields to be ETFs, including new criteria involving triple products and connections to combinatorial design theory.

Findings

Necessary and sufficient conditions for ETFs over finite fields.

Extension of Welch bound saturation criteria.

Connections between equiangular lines and combinatorial designs.

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.