Asymptotic Estimates of the Probability of Error for Transmission of Messages over a Discrete Memoryless Communication Channel with a Symmetric Transition Probability Matrix

A memoryless channel with a matrix of transaction probabilities $P=\{P_{ij}\}$ is considered such that any row of the matrix $P$ is a permutation of any other row and any column is a permutation of any other column. It is supposed that $2^{nH}$ possible messages are to be transmitted with the help of words consisting of $n$ symbols. The asymptotic behavior of the error probability for the optimal code is investigated as is the asymptotic behavior of the expectation of the error probability of the randomly chosen code.