Counting Problems for Orthogonal Sets and Sublattices in Function Fields

Abstract

Let $\mathcal{K}=\mathbb{F}_q((x^{-1}))$. Analogous to orthogonality in the Euclidean space $\mathbb{R}^n$, there exists a well-studied notion of ultrametric orthogonality in $\mathcal{K}^n$. In this paper, we extend the work of Soffer-Aranov and Behajaina on counting problems related to orthogonality in $\mathcal{K}^n$. For example, we resolve an open question posed in Soffer-Aranov and Behajaina by bounding the size of the largest ``orthogonal sets'' in $\mathcal{K}^n$. Furthermore, using similar ideas and techniques, we investigate analogues of Hadamard matrices over $\mathcal{K}$. Finally, we also use ultrametric orthogonality to compute the number of sublattices of $\mathbb{F}_q[x]^n$ with a certain geometric structure, and to determine the number of orthogonal bases of a sublattice in $\mathcal{K}^n$. The resulting formulas depend crucially on successive minima.