Rubinstein’s exact quadratic separation

For every natural number $m$ with $m\ge 1$, there exists a Boolean function $f:\{0,1\}^{(2m)^2}\to\{0,1\}$ with the following two exact equalities. For an input $x$ and a set $B$ of coordinate indices, let $x^B$ be the input obtained by complementing exactly the coordinates in $B$. Define $s(f)$ to be the maximum, over all inputs $x$, of the number of indices $i$ for which $f(x^{\{i\}})\ne f(x)$. Define $b(f)$ to be the maximum, over all inputs $x$, of the largest cardinality of a family of distinct, pairwise disjoint, nonempty sets $B$ of coordinate indices such that $f(x^B)\ne f(x)$ for every member $B$ of the family; the empty family is allowed and has cardinality zero. Then $s(f)=2m$ and $b(f)=2m^2$, where the latter means $2\cdot(m^2)$, whereas the number of coordinates is $(2m)^2=4m^2$. Both maxima are taken over all Boolean inputs independently; the statement does not require the same input to attain them. The function may depend on $m$ and is not asserted to be unique. The case $m=1$ is included, and $m=0$ is excluded by the hypothesis.