Consider the set $mathcal{A}_n$ of all $n times n$ binary matrices in which
the number of $1$'s in each row and column is at most $n/2$. We show that the
redundancy, $n^2 - log_2|mathcal{A}_n|$, of this set equals $rho n - delta
sqrt{n} + O(log n)$, for a constant $rho approx 1.42515$, and $delta =
delta(n) approx 1.46016$ for even $n$ and $0$ otherwise.