Combinatorics of Euclidean spaces over finite fields

The $q$-binomial coefficients are q-analogues of the binomial coefficients, counting the number of $k$-dimensional subspaces in the $n$-dimensional vector space $\mathbb{F}^n_q$ over $\mathbb{F}_{q}$. In this paper, we define a Euclidean analogue of $q$-binomial coefficients as the number of $k$-dimensional subspaces which have an orthonormal basis in the quadratic space $(\mathbb{F}_{q}^{n},x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})$ using a poset structure on these subspaces. We prove its various combinatorial properties comparing with those of $q$-binomial coefficients. In addition, we formulate the number of subspaces of other quadratic types and study some related properties.