Counting Problems for Orthogonal Sets and Sublattices in Function Fields

TL;DR

This paper extends counting problems related to ultrametric orthogonality in function fields, providing bounds, analogues of Hadamard matrices, and formulas for sublattices and orthogonal bases, advancing understanding in algebraic combinatorics.

Contribution

It introduces new bounds and formulas for orthogonal sets, sublattices, and orthogonal bases in function fields, addressing open questions and developing analogues of classical matrices.

Findings

Bound on the size of largest orthogonal sets in $\\mathcal{K}^n$

Formulas for counting sublattices with specific structures

Characterization of orthogonal bases in ultrametric spaces

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.