Finite-Blocklength Per-User Error Bounds and Matched-Poisson Decoding for Unsourced Optical Access

Abstract

Unsourced random access (URA) lets many users share one codebook, with the receiver returning an unordered message list under a per-user probability of error (PUPE) criterion. It has been developed primarily for Gaussian and fading channels. This letter formulates URA for the photon-limited optical regime: a Poisson intensity-modulation/direct-detection (IM/DD) multiple-access channel under peak and ensemble-average intensity constraints. We derive a finite-blocklength PUPE achievability bound for a common on–off codebook that counts the overlap between the true and competing message lists exactly, and this bound yields a deterministic required-peak design rule. At target PUPE 0.05, reliability is certified at received peaks below one mean detected count per channel use for up to one hundred active users at low background. A list-Fano necessary condition connects the bound to the exact peak-constrained Lapidoth–Shamai sum-rate reference; it is never contradicted across a $6{,}600$ -point grid, and the common $a_{\mathrm {peak}}/2$ proxy understates the exact ceiling by up to 13.8% at low background. Finally, small exact-enumerable comparisons isolate the channel law: replacing the Euclidean rule by the matched Poisson likelihood lowers PUPE by a mean of 72% per cell at zero background. This advantage vanishes as background counts dominate the shot-noise variation.