On the rank of random matrix over prime field consisting of independent rows with given numbers of nonzero elements

In a recent paper we had proposed explicit bound for the distribution function of the rank of matrix with independent rows having fixed weights. Here this bound is generalized for a wider class of binary matrices with independent rows and also to matrices over prime field ${GF}(p)$ that consist of independent rows, which are chosen from sets of vectors with given numbers of non-zero elements.