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

Abstract

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.