Range dependence of pulse position modulation in the presence of background noise

We identify the maximum attainable transmission rate of a noisy optical link employing the pulse position modulation (PPM) format with direct detection. We show that for a fixed background noise level it is possible to achieve the information rate directly proportional to the average detected signal power in the photon-starved regime. This implies inverse-square scaling with the distance, presenting a qualitative improvement over previously obtained estimates that scaled as the inverse of the fourth power of the distance. The necessary ingredients to achieve the improved mode of operation are the unrestricted optimization of the PPM order and the complete decoding of detection events to extract information from sequences containing multiple counts within one PPM frame. Importantly, information efficiency equivalent to high-order PPM formats can be attained using signals with uniformly distributed optical power processed with recently proposed structured optical receivers.