The Maximum Error Probability Criterion, Random Encoder, and Feedback, in Multiple Input Channels

For a multiple input channel, one may define different capacity regions, according to the criterions of error, types of codes, and presence of feedback. In this paper, we aim to draw a complete picture of relations among these different capacity regions. To this end, we first prove that the average-error-probability capacity region of a multiple input channel can be achieved by a random code under the criterion of maximum error probability. Moreover, we show that for a non-deterministic multiple input channel with feedback, the capacity regions are the same under two different error criterions. In addition, we discuss two special classes of channels to shed light on the relation of different capacity regions. In particular, to illustrate the roles of feedback, we provide a class of MAC, for which feedback may enlarge maximum-error-probability capacity regions, but not average-error-capacity regions. Besides, we present a class of MAC, as an example for which the maximum-error-probability capacity regions are strictly smaller than the average-error-probability capacity regions (first example showing this was due to G. Dueck). Differently from G. Dueck’s enlightening example in which a deterministic MAC was considered, our example includes and further generalizes G. Dueck’s example by taking both deterministic and non-deterministic MACs into account. Finally, we extend our results for a discrete memoryless two-input channel, to compound, arbitrarily varying MAC, and MAC with more than two inputs.