On coding for the ideal Poisson channel with noiseless feedback

The ideal Poisson channel with noiseless feedback models a direct detection photon channel which does not suffer from dark current, and where a causal feedback link informs the transmitter at time t of the channel output at all times prior to t. A coding scheme for the channel 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. It is shown that under a capacity reducing average power constraint, the upper bound coincides with the error exponent achieved by the proposed coding scheme. This provides a nontrivial example of a channel for which the error exponent is known exactly both with and without feedback