# Orthogonal Matrices and Cosets (translates) of Linear Subspaces

Let $$M_n(F_2)$$ be the vector space of all $$n\times n$$ matrices over the finite field $$F_2$$. Let $$O(n)\subset M_n(F_2)$$ be the set of all orthogonal matrices and $$W\subseteq O(n)$$ be an affine subspace (coset of a linear subspace) of dimension $$k$$. What is the largest possible value of $$k$$? That is I want to find the largest size of affine subspaces in $$O(n)$$.

• Since |W| = 2^dim(W), and the 2-part of |O(n)| is 2^([n/2]^2), it follows that k is at most [n/2]^2. This bound is attained if n is 2 or 3, but even for n=4 I only see how to create a 2-dimensional W. – user38495 Jan 8 at 4:04