Finite-Blocklength Results for the A-channel: Applications to Unsourced Random Access and Group Testing

TL;DR

This paper derives finite-blocklength bounds for the A-channel, demonstrating its applications in unsourced random access and group testing, and proposing improved decoding and testing matrix constructions.

Contribution

It introduces finite-blocklength achievability bounds for the A-channel and connects these results to enhanced group testing and random access methods.

Findings

Achieves bounds for error probabilities in group testing with A-channel codes.

Provides constructions of group testing matrices with sub-linear decoding time.

Shows asymptotic optimality in the number of tests for certain group testing scenarios.

Abstract

We present finite-blocklength achievability bounds for the unsourced A-channel. In this multiple-access channel, users noiselessly transmit codewords picked from a common codebook with entries generated from a $q$-ary alphabet. At each channel use, the receiver observes the set of different transmitted symbols but not their multiplicity. We show that the A-channel finds applications in unsourced random-access (URA) and group testing. Leveraging the insights provided by the finite-blocklength bounds and the connection between URA and non-adaptive group testing through the A-channel, we propose improved decoding methods for state-of-the-art A-channel codes and we showcase how A-channel codes provide a new class of structured group testing matrices. The developed bounds allow to evaluate the achievable error probabilities of group testing matrices based on random A-channel codes for arbitrary numbers of tests, items and defectives. We show that such a construction asymptotically achieves the optimal number of tests. In addition, every efficiently decodable A-channel code can be used to construct a group testing matrix with sub-linear recovery time.