Bounds on the Average Error Probability for a Code Ensemble with Fixed Composition

The author obtains the logarithmic asymptotic form of the average error probability over a code ensemble with fixed composition [R. M. Fano,Transmission of Information: A Statistical Theory of Communications, New York, MIT Press, 1961] for the case of decoding by a list of fixed length $l\geq 1$, for a discrete memoryless channel. The exponent of the error probability is represented in an analytic form analogous to that of the exponent of the spherical packing bound for codes with fixed composition introduced by E. A. Haroutunian in [Probl. Peredachi Inf., 1968, vol. 4, no. 4, pp. 37–48]. In the particular case $L=1$, corresponding to classical decoding this exponent coincides with the exponent of the upper bound proved by Fano.