A limit theorem for a characteristic of a random Boolean matrix

Let $\|a_i^j,\ i=1,\dots,k,\ j=1,\dots,n\|$, $k=[n\theta]$, $0<\theta<1$, be a Boolean matrix with mutually independent random elements $a_i^j$:
$$ \mathbf P\{a_i^j=1\}=\pi_i^j,\quad0<\pi_i^j<1. $$

We consider the minimum distance $\zeta$ of a random linear code with parity-check matrix $\|a_i^j\|$.
Theorem 1. {\it Let all $\pi_i^j\in[\delta,1-\delta]$ where $\delta$ is a fixed positive number. Then {(3)} holds uniformly for $\pi_i^j\in[\delta,1-\delta]$ and for $t$ subject to} (1), (2).
Theorem 2. (3) holds uniformly for $\pi_i^j\in[\delta_n,1-\delta_n]$ as $\delta_n\to0$, $\delta_nn/\ln n\to\infty$ and for $t$ subject to (4), (5).