  The existence of unimodular forms with small norms on sequence spaces is
crucial in a variety of problems in modern analysis. We prove that the infimum
of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real)
forms $A$ on $\ell_{p_1}^{n_{1}} \times \cdots \times \ell_{p_d}^{n_{d}}$, for
all $p_1,\dots,p_d \in [2, \infty]$ and all positive integers $n_1,\dots,n_d$,
behaves (asymptotically) as $\left(n_{1}^{1/2} + \cdots + n_{d}^{1/2}\right)
\prod_{j=1}^{d}n_{j}^{\frac{1}{2} - \frac{1}{p_j}}$. Applications to the theory
of the multilinear Hardy--Littlewood inequality are also presented.
