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. As in the model recently introduced by Polyanskiy for the Multiple Access Channel (MAC) with a fixed, known number of transmitters, the channel is assumed to be invariant to permutations on its inputs, and all active transmitters employ identical encoders. Unlike the Polyanskiy model, in the proposed scenario, neither the transmitters nor the receiver knows which transmitters are active. We refer to this agnostic communication setup as the Random Access Channel (RAC). Scheduled feedback of a finite number of bits is used to synchronize the transmitters. 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, k, 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.Index Terms-Channel coding, random access channel, finite blocklength regime, achievability, second-order asymptotics, rateless codes. I. INTRODUCTIONAccess points like WiFi hot spots and cellular base stations are, for wireless devices, the gateway to the network. Unfortunately, access points are also the network's most critical bottleneck. As more kinds of devices become network-reliant, both the number of communicating devices and the diversity of their communication needs grow. Little is known about how to code under high variation in the number and variety of communicators.Multiple-transmitter single-receiver channels are well understood in information theory when the number and identities of transmitters are fixed and known. Unfortunately, even in this known-transmitter regime, information-theoretic solutions are too complex to implement. As a result, orthogonalization methods, such as TDMA, FDMA, and orthogonal CDMA, are

Variable-length Feedback Codes with Several Decoding Times for the Gaussian Channel

“…In this section, we show that the error value N chosen in (123) is second-order optimal in the sense that under the VLSF code design in Section IV-D, the error value * N that minimizes the average decoding time in (126) has the same asymptotic expansion (19) of the maximum achievable message size as N up to the second-order term.…”

“…Random variable U is common randomness shared by the transmitter and receiver. As in [6], [13], [19], the traditional 2 The realization u of U specifies the codebook. random-coding argument does not prove the existence of a single (deterministic) code that simultaneously satisfies two conditions on the code (e.g., ( 9) and ( 10)).…”

“…are shown for K = 1 (14) and K ∈ {2, 3, 4} (Theorem 1), P = 1 and = 10 −3 . The achievability bounds use the asymptotic approximation, i.e., we ignore the O(•) term in (19).…”

“…When K < ∞ and the probability P [ı(X n K ; Y n K ) < γ] dominates, using i.i.d. N (0, P ) inputs achieves a worse second-order term in the asymptotic expansion (19) of the maximum achievable message size. This implies that when K < ∞, using our uniform distribution on a subset of the power sphere is superior to choosing codewords i.i.d.…”

“…In particular, i.i.d. N (0, P ) inputs achieve (19), where the dispersion V (P ) is replaced by the variancẽ V (P ) = P 1+P of ı(X; Y ) when X ∼ N (0, P ); hereṼ (P ) is greater than the dispersion V (P ) for all P > 0 (see [25, eq. (2.56)]).…”