On probability of correction of a random number of errors in an error-correcting coding

We consider the probability $\mathbf P(A)$ of the event $A$ that while $n$ messages each consisting of $N$ blocks are encoded by a Hamming-type code all errors are corrected. It is assumed that the ith message has $m_i=m_i(\omega_1)$ errors, $\omega_1\in\Omega_1$, where $m_i$ are independent identically distributed random variables defined on the probability space $(\Omega_1,\mathfrak A_1,\mathbf P _1)$. The probability $\mathbf P(A)$ is determined in the framework of the generalised allocation scheme introduced by V. F. Kolchin. It is shown that in the case where $n,N\to\infty$ in such a manner that $\alpha=n/N\to\alpha_0<\infty$ the probabilities $\mathbf P(A)$ converge to one and the same limit for almost all $\omega_1\in\Omega_1$, and the value of this limit is found.