  We analyze the information efficiency of a deep-space optical communication
link with background noise employing the pulse position modulation (PPM) format
and a direct-detection receiver based on Geiger-mode photon counting. The
efficiency, quantified using Shannon mutual information, is optimized with
respect to the PPM order under the constraint of a given average signal power
in simple and complete decoding scenarios. We show that the use of complete
decoding, which retrieves information from all combinations of detector
photocounts occurring within one PPM frame, allows one to achieve information
efficiency scaling as the inverse of the square of the distance, i.e.
proportional to the received signal power. This represents a qualitative
enhancement compared to simple decoding, which treats multiple photocounts
within a single PPM frame as erasures and leads to inverse-quartic scaling with
the distance. We provide easily computable formulas for the link performance in
the limit of diminishing signal power.
