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 $O(n)$.

  • $\begingroup$ 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. $\endgroup$ – user38495 Jan 8 at 4:04

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