  The Kahane--Salem--Zygmund inequality is a probabilistic result that
guarantees the existence of special matrices with entries $1$ and $-1$
generating unimodular $m$-linear forms $A_{m,n}:\ell_{p_{1}}^{n}\times
\cdots\times\ell_{p_{m}}^{n}\longrightarrow\mathbb{R}$ (or $\mathbb{C}$) with
relatively small norms. The optimal asymptotic estimates for the smallest
possible norms of $A_{m,n}$ when $\left\{ p_{1},...,p_{m}\right\}
\subset\lbrack2,\infty]$ and when $\left\{ p_{1},...,p_{m}\right\}
\subset\lbrack1,2)$ are well-known and in this paper we obtain the optimal
asymptotic estimates for the remaining case: $\left\{ p_{1},...,p_{m}\right\} $
intercepts both $[2,\infty]$ and $[1,2)$. In particular we prove that a
conjecture posed by Albuquerque and Rezende is false and, using a special type
of matrices that dates back to the works of Toeplitz, we also answer a problem
posed by the same authors.
