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We show that with high probability, random rank 1 matrices over a finitefield are in (linearly) general position, at least provided their shape k x lis not excessively unbalanced. This translates into saying that the dimensionof 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 ofCascudo-Cramer-Mirandola-Zemor dealing with *-squares of codes, which itcomplements, especially regarding applications to the analysis of McEliece-typecryptosystems. We also briefly mention the case of higher *-powers, whichrequire to take the Frobenius into account. We then conclude with some openproblems.

Linear independence of rank 1 matrices and the dimension of *-products of codes