  We show that with high probability, random rank 1 matrices over a finite
field are in (linearly) general position, at least provided their shape k x l
is not excessively unbalanced. This translates into saying that the dimension
of the *-product of two [n, k] and [n, l] random codes is equal to min(n, kl),
as one would have expected. Our work is inspired by a similar result of
Cascudo-Cramer-Mirandola-Zemor dealing with *-squares of codes, which it
complements, especially regarding applications to the analysis of McEliece-type
cryptosystems. We also briefly mention the case of higher *-powers, which
require to take the Frobenius into account. We then conclude with some open
problems.
