An Exploration of a New Group of Channel Symmetries

We study a certain symmetry group associated to any given communication channel, which can informally be viewed as the set of transformations of the set of inputs that "commute" with the action of the channel. In a general setting, we show that the distribution over the inputs that maximizes the mutual information between the input and output of a given channel is a "fixed point" of the action of the channel's group. We consider as examples the groups associated with the binary symmetric channel and the binary deletion channel. We show that the group of the binary symmetric channel is extremely large (it contains a number of elements that grows faster than any exponential function of $n$), and we give empirical evidence that the group of the binary deletion channel is extremely small (it contains a number of elements constant in $n$). This serves as some formal justification for why the analysis of the binary deletion channel has proved much more difficult than its memoryless counterparts.