Infinite families of optimal systems of biangular lines related to representations of $\textrm{SL}(2,\mathbb{F}_q)$

TL;DR

This paper introduces two infinite families of optimal biangular line packings derived from representations of SL(2, F_q), achieving known bounds and contributing to the theory of line packings, mutually unbiased bases, and projective 2-designs.

Contribution

It presents new infinite families of biangular line packings that meet the second Levenshtein bounds, constructed via representations of SL(2, F_q), and relates them to mutually unbiased weighing matrices and projective 2-designs.

Findings

New infinite biangular line packings meet second Levenshtein bounds.

Packings are related to maximal sets of mutually unbiased weighing matrices.

Packings are projective 2-designs, improving known bounds on biangular tight frames.

Abstract

A line packing is optimal if its coherence is as small as possible. Most interesting examples of optimal line packings are achieving equality in some of the known lower bounds for coherence. In this paper two infinite families of real and complex biangular line packings are presented. New packings achieve equality in the real or complex second Levenshtein bound respectively. Both infinite families are constructed by analyzing well known representations of the finite groups SL$(2,\mathbb{F}_q)$. Until now the only known infinite familes meeting the second Levenshtein bounds were related to the maximal sets of mutually unbiased bases (MUB). Similarly to the line packings related to the maximal sets of MUBs, the line packings presented here are related to the maximal sets of mutually unbiased weighing matrices. Another similarity is that the new packings are projective 2-designs. The latter property together with sufficiently large cardinalities of the new packings implies some improvement on largest known cardinalities of real and complex biangular tight frames.