  An n\times n matrix M is called a fooling-set matrix of size n, if its
diagonal entries are nonzero, whereas for every k\ne \ell we have M_{k,\ell}
M_{\ell,k} = 0. Dietzfelbinger, Hromkovi\v{c}, and Schnitger (1996) showed that
n \le (\rk M)^2, regardless of over which field the rank is computed, and asked
whether the exponent on \rk M can be improved.
  We settle this question for nonzero characteristic by constructing a family
of matrices for which the bound is asymptotically tight. The construction uses
linear recurring sequences.
