On binary channels and their cascades

A detailed analysis of the general binary channel is given, with special reference to capacity (both separately and in cascade), input and output symbol distributions, and probability of error. The infinite number of binary channels with the same capacity lie on double-branched equicapacity lines. Of the channels on the lower branch of a given equicapacity line, the symmetric channel has the smallest probability of error and the largest capacity in cascade, unless the capacity is small, in which case the asymmetric channel (with one noiseless symbol) has the smallest probability of error and the largest capacity in cascade. By simply reversing the designation of the output (or input) symbols, we can decrease the probability of error of any channel on the upper branch of the equi-capacity line and increase the capacity in cascade of any asymmetric channel on the upper branch. In a binary channel neither symbol should be transmitted with a probability lying outside the interval
[1/e, 1 - (1/e)]
if capacity is to be achieved. The maximally asymmetric input symbol distributions are approached by certain low-capacity channels. For these channels, redundancy coding permits an appreciable fraction of capacity in cascade if sufficient delay can be tolerated.