On the Maximum Probability of Full Rank of Random Matrices over Finite Fields

The problem of determining the conditions under which a random rectangular matrix is of full rank is a fundamental question in random matrix theory, with significant implications for coding theory, cryptography, and combinatorics. In this paper, we study the probability of full rank for a K×N random matrix over the finite field Fq, where q is a prime power, under the assumption that the rows of the matrix are sampled independently from a probability distribution P over FqN. We demonstrate that the probability of full rank attains a local maximum when the distribution P is uniform over FqN∖{0}, for any K⩽N and prime power q. Moreover, we establish that this local maximum is also a global maximum in the special case where K=2. These results highlight the optimality of the uniform distribution in maximizing full rank and represent a significant step toward solving the broader problem of maximizing the probability of full rank for random matrices over finite fields.