Random Access Channel Coding in the Finite Blocklength Regime

Abstract

Consider a random access communication scenario over a channel whose operation is defined for any number of possible transmitters. Inspired by the model recently introduced by Polyanskiy for the Multiple Access Channel (MAC) with a fixed, known number of transmitters, we assume that the channel is invariant to permutations on its inputs, and that all active transmitters employ identical encoders. Unlike Polyanskiy, we consider a scenario where neither the transmitters nor the receiver know which transmitters are active. We refer to this agnostic communication setup as the Random Access Channel, or RAC. Limited feedback is used to ensure that the collection of active transmitters remains fixed during each epoch. The decoder is tasked with determining from the channel output the number of active transmitters (k) and their messages but not which transmitter sent which message. The decoding procedure occurs at a time nt depending on the decoder's estimate t of the number of active transmitters, thereby achieving a rate that varies with the number of active transmitters. Single-bit feedback at each time ni, i ≤ t, enables all transmitters to determine the end of one coding epoch and the start of the next. The central result of this work demonstrates the achievability on a RAC of performance that is first-order optimal for the MAC in operation during each coding epoch. While prior multiple access schemes for a fixed number of transmitters require 2 k − 1 simultaneous threshold rules, the proposed scheme uses a single threshold rule and achieves the same dispersion.

Lossless Source Coding in the Point-to-Point, Multiple Access, and Random Access Scenarios

“…This strategy enables us to accommodate an arbitrarily large number of encoders without the burden of designing a unique encoding map for each. A similar phenomenon arises for RA channel coding, as previously studied in [17].…”

Gaussian Multiple and Random Access in the Finite Blocklength Regime

“…This converse does not match the second-order term in the achievability bounds proven in this paper. The gap in the second-order analyses of current MAC achievability and converse results is a challenging open problem, as discussed in [25].…”

“…Communication strategy: We adapt the epoch-based rateless communication strategy we put forth in [20] to achieve the fundamental limits of the Gaussian RAC. Each transmitter is either active or silent during a whole epoch.…”

“…In [20], we develop a communication strategy for a general RAC where neither the transmitters nor the receiver knows the set of active transmitters. A central result of that work shows that for permutation-invariant RACs, under mild conditions it is possible to achieve performance identical in the firstand second-order terms to the best performance known to be achievable for the underlying MAC.…”

“…In [21], Liu and Effros achieve improved thirdorder bounds using a maximum-likelihood decoder. Although the coding strategies proposed in [20], [21] apply to the Gaussian RAC, the random encoder design in [20] uses an i.i.d. input distribution.…”