On establishing the Shannon ordering for discrete memoryless channels

This paper studies several problems concerning channel inclusion, which is a partial ordering between discrete memoryless channels (DMCs) proposed by Shannon. Specifically, checkable conditions are derived for channel inclusion between DMCs with certain special structure, and these conditions are related to the mathematical theory of majorization. The conditions for channel inclusion between binary erasure channel (BEC) and binary symmetric channel (BSC) are also derived, which are shown to be equivalent to those for channel output degradation. Furthermore, the determination of channel inclusion is considered as a convex optimization problem, and the sparsity of the weights related to the representation of the worse DMC in terms of the better one is revealed when channel inclusion holds between two DMCs.