ε-Capacity of Binary Symmetric Averaged Channels

We consider the channel model obtained by averaging binary symmetric channel (BSC) components with respect to a weighting distribution. A nonempty open interval (A, B) is called a capacity gap for this channel model if no channel component has capacity in (A, B) and this property fails for every open interval strictly containing (A, B). For a fixed epsi>0, suppose one wishes to compute the epsi-capacity of the channel, which is the maximum asymptotic rate at which the channel can be encoded via a sequence of channel codes each achieving block error probability lesepsi. In 1963, Parthasarathy provided a formula for epsi-capacity which is valid for all but at most countably many values of epsi. When the formula fails, there exists a unique capacity gap (A, B) such that the epsi-capacity lies in [A, B], but one does not know precisely where. Via a coding theorem and converse, we establish a formula for computing epsi-capacity as a function of the endpoints A, B of the associated capacity gap (A, B); the formula holds whenever the capacity gap is sufficiently narrow in width