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

TL;DR

This paper proves that random rank 1 matrices are in general position with high probability, leading to the expected dimension of *-products of random codes, with implications for cryptosystems and higher *-powers.

Contribution

It establishes the high-probability linear independence of random rank 1 matrices and determines the dimension of *-products of random codes, extending previous results and applications.

Findings

Random rank 1 matrices are in general position with high probability.

Dimension of *-product of two random codes is min(n, kl).

Results have implications for cryptosystems like McEliece.

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.