  Given a linear code $C$, one can define the $d$-th power of $C$ as the span
of all componentwise products of $d$ elements of $C$. A power of $C$ may
quickly fill the whole space. Our purpose is to answer the following question:
does the square of a code "typically" fill the whole space? We give a positive
answer, for codes of dimension $k$ and length roughly $\frac{1}{2}k^2$ or
smaller. Moreover, the convergence speed is exponential if the difference
$k(k+1)/2-n$ is at least linear in $k$. The proof uses random coding and
combinatorial arguments, together with algebraic tools involving the precise
computation of the number of quadratic forms of a given rank, and the number of
their zeros.
