Range dependence of an optical pulse position modulation link in the presence of background noise

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.