On the reliability function of the ideal Poisson channel with noiseless feedback

A coding scheme for the channel under peak power and average power constraints on the input is presented, and its asymptotic error exponent is shown to coincide, at all rates below capacity, with the sphere packing error exponent, which, for the case at hand, is known to be unachievable without feedback for rates below the critical rate. An upper bound on the error exponent achievable with feedback is also derived and shown, under a capacity reducing average power constraint, to coincide with the error exponent achieved by the proposed coding scheme; in such a case the coding scheme is asymptotically optimal. Thus, the ideal Poisson channel, limited by a capacity-reducing average power constraint, provides a nontrivial example of a channel for which the reliability function is known exactly both with and without feedback. It is shown that a slight modification of the coding scheme to one of random transmission time can achieve zero-error probability for any rate lower than the ordinary average-error channel capacity