  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.
