Full-rank conjecture for regular and irregular banded random binary matrices

Let $n,k,\alpha$ and $\gamma$ be integers with $k\leq n$ and $\gamma<\alpha$. Let $M$ be a $(\gamma,\alpha)$ regular or irregular symmetric or asymmetric banded random matrix of size $n\times k$. Its rank over the binary field is denoted by $r(M)$; the parameters satisfy $\gamma=\alpha/\tau_e$ and $\tau_e=\tau/(\tau-1)$, where $\tau$ is any constant divisor of $\alpha$. Full-rank conjecture. For any $\epsilon>0$ and sufficiently large $k$,

$$\Pr[r(M)<k]\leq\epsilon,$$

provided $k\leq n-\log(1/\epsilon)$ and either $\gamma\geq 2\sqrt{k}$ or $\gamma\geq\tau_e\tau\sqrt{k}$, respectively. This conjecture concerns the rank behavior of two general classes of banded random binary matrices, extending a prior result for a subclass of symmetric banded matrices; its resolution would establish that these matrices are full rank with high probability under the stated overlap conditions.