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 ${\mathbb{F}}_q^n$ 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.