The Probability of Error in Block Transmission in a Memoryless Gaussian Channel with Feedback

Block methods of transmission in a stationary, memoryless Gaussian channel with complete feedback are described and investigated. It is shown that, for any transmission rate $R$, $0<R<C$, where $C$ is the channel capacity, the optimum probability of error coincides, to within a factor, with the lower boundaries for the probability of error obtained by Shannon [Bell Syst. Tech. J., 1959, vol. 38, p. 611] for discrete time and Fano [R. M. Fano, Transmission of Information, New York, M.I.T. Press, 1961] for continuous time. For $R\to 0$ as $\tau\to\infty$ a method of transmission is constructed such that the index of the exponent of the probability of error coincides with the index of the exponent of the probability of error for the transmission of two signals.