p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound

Abstract

We introduce the notion of p-adic equiangular lines and derive the first fundamental relation between common angle, dimension of the space and the number of lines. More precisely, we show that if $\{\ddot{I}{„}_j{\}}_{j=1}^n$ is p-adic ${\hat{I}}^3$-equiangular lines in ${\mathbb{Q}}_p^d$, then \begin{align*} (1) \quad\quad \quad \quad |n|^2\leq |d|\max\{|n|, γ^2 \}. \end{align*} We call Inequality (1) as the p-adic van Lint-Seidel relative bound. We believe that this complements fundamental van Lint-Seidel \textit{[Indag. Math., 1966]} relative bound for equiangular lines in the p-adic case.

7 Pages, 0 Figures. p-adic Gerzon bound is added