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A direct-detection photon-limited optical communication channel that uses pulse position modulation (PPM) under a pulsewidth constraint is considered. Overlapping PPM (OPPM) allows multiple positions per pulsewidth, as well as fractional modulation indices (number of pulsewidths per frame) requiring more refined timing than that needed for conventional disjoint PPM (DJPPM). It is shown that even at moderate values of the expected photon count per pulse --such as needed for high data rates--OPPM outperforms on-off keying (OOK) in both capacity and cutoff rate, even though OOK is uniformly superior to DJPPM. Moreover, efficient use of OPPM is possible with equiprobable input symbols, whereas OOK requires inconvenient asymmetrical inputs to achieve capacity and high cutoff rate efficiencies (nats/ photon). At lower data rates, where capacity efficiency is the prime criterion, a significant advantage (- 20 percent) over DJPPM can be achieved up to efficiencies of about 0.7 nats/ photon. The M-ary photon-limited OPPM channel can be viewed as an ambiguity and erasure channel, in the sense that some channel outputs are ambiguous in only some input symbols and only if no photons are counted is there ambiguity in all input symbols. For large ambiguities cause bursts of erasures of data symbols. Massey's interlaced encoding, as well as conventional encoding followed by interleaving, are adaptable to this bursty channel, and effect an increase in its cutoff rate comparable to the increase obtainable with DJPPM by the same techniques.