Combinatorics of Euclidean spaces over finite fields

Let $\mathbb{F}_{q}$ be the finite field with an odd prime power $q$. In this paper, we study various combinatorial properties related to non-degenerate quadratic spaces over finite fields. First, we investigate the Euclidean poset $E_{n}(q)$, which consists of all subspaces of $(\mathbb{F}_{q}^{n},\text{Euc}_{n})$ that have an orthonormal basis, where $\text{Euc}_{n}(\mathbf{x}):=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}$. Using this poset structure, we show that the number of $k$-dimensional subspaces of $(\mathbb{F}_{q}^{n},\text{Euc}_{n})$ that have an orthonormal basis behaves like the binomial coefficient, which we call the Euclidean-binomial coefficient $\binom{n}{k}_{q}^{\perp}$ for $k=1,\cdots,n$. The main purpose of this paper is to study its various combinatorial properties.Comment: 21 pages, comments are welcom