Exact Asymptotics of Random Coding Error Probability for General Memoryless Channels

Junya Honda

Abstract: This paper considers the error probability of random coding for memoryless channels in the fixed-rate regime. Random coding is a fundamental tool to construct a capacity-achieving channel code and many studies have been conducted for the asymptotics of the decoding error probability. Gallager derived the exact asymptotics (that is, an evaluation with asymptotically vanishing relative error) of the error probability below the critical rate. On the other hand, the asymptotics for the rate above the critical rate has been unknown except for symmetric channels (in the strong sense) and strongly nonlattice channels. This paper derives the exact asymptotics for general memoryless channels covering all previously unsolved cases. The analysis reveals that strongly symmetric channels and strongly nonlattice channels correspond to two extreme cases and the expression of the asymptotics is much complicated for general channels.