The smallest singular value of products of random combinatorial and deterministic matrices

Let $p\in(0,1/2]$ be fixed, and let $d,n$ be positive integers satisfying $d=pn$. Let $Q_n$ be an $n\times n$ random combinatorial matrix whose rows are independent and uniformly distributed over the vectors in $\{0,1\}^n$ having exactly $d$ ones. For an arbitrary deterministic invertible $n\times n