Counting Problems for Orthogonal Sets and Sublattices in Function Fields

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.

💡 Research Summary

The paper studies orthogonality, Hadamard matrices, and sublattice structures in the ultrametric setting of the function field 𝔎 = 𝔽_q((x⁻¹)).
First, the authors adopt the ultrametric norm ‖·‖ on 𝔎ⁿ (‖v‖ = max_i |v_i|) and define two notions of orthogonality: a set {u₁,…,u_ℓ} is orthogonal if for every choice of scalars λ_i we have ‖∑λ_i u_i‖ = max_i‖λ_i u_i‖, while weak orthogonality only requires pairwise orthogonal vectors. Using the reduction map γ_n : B_n → 𝔽_qⁿ (modulo the maximal ideal) and the projective map ρ_n : 𝔽_qⁿ → ℙ^{n‑1}(𝔽_q), they prove (Lemma 2.2) that a subset of the unit sphere B_n is orthogonal precisely when its image under γ_n is linearly independent over 𝔽_q. Consequently, (k,ℓ)-orthogonal sets correspond to subsets whose reduction spans a projective subspace of dimension at least ℓ‑1 (Lemma 2.3).

With this correspondence, the authors obtain sharp bounds for the maximal size Θ(k,ℓ)_n of a (k,ℓ)-orthogonal set (Theorem 1.6). The upper bound is (k‑1)(q^ℓ‑1)/(q‑1)·(qⁿ‑1), and it is attained whenever (q^ℓ‑1)/(q‑1) divides k‑1. They also show that Θ(k,ℓ)_n = Δ(k,ℓ)_n (the weak version) if and only if ℓ = 2, resolving Question 3.28 from their earlier work. The extremal configurations are completely described in Theorem 1.8: when ℓ‑1 | k‑1, each projective point in ℙ^{n‑1}(𝔽_q) contains exactly (k‑1)/(ℓ‑1) elements of a maximal weakly orthogonal set; when (q^ℓ‑1)/(q‑1) | k‑1, each point contains (k‑1)(q‑1)q^{ℓ‑1}/(q‑1) elements of a maximal orthogonal set.

The second major contribution is the definition and enumeration of Hadamard matrices over the polynomial ring R = 𝔽_q