Asymmetric SICs over finite fields

Abstract

Zauner's conjecture concerns the existence of $d^2$ equiangular lines in ${\mathbb{C}}^d$; such a system of lines is known as a SIC. In this paper, we construct infinitely many new SICs over finite fields. While all previously known SICs exhibit Weyl--Heisenberg symmetry, some of our new SICs exhibit trivial automorphism groups. We conjecture that such \textit{totally asymmetric} SICs exist in infinitely many dimensions in the finite field setting.