Improved lower bounds on the rigidity of Hadamard matrices

We write$f=\Omega(g)$ if $f(x)\ge cg(x)$ with some positive constant $c$ for all $x$ from the domain of functions $f$ and $g$. We show that at least $\Omega(n^2/r)$ entries must be changed in an arbitrary (generalized) Hadamard matrix in order to reduce its rank below $r$. This improves the previously known bound $\Omega(n^2/r^2)$. If we additionally know that the changes are bounded above in absolute value by some number $\theta\ge n/r$, then the number of these entries is bounded below by $\Omega(n^3/(r\theta^2))$, which improves upon the previously known bound $\Omega(n^2/\theta^2)$.