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Unsourced random-access (U-RA) is a type of grant-free random access with a virtually unlimited number of users, of which only a certain number K<inf>a</inf> are active on the same time slot. Users employ exactly the same codebook, and the task of the receiver is to decode the list of transmitted messages. Recently a concatenated coding construction for U-RA on the AWGN channel was presented, in which a sparse regression code (SPARC) is used as an inner code to create an effective outer OR-channel. Then an outer code is used to resolve the multiple-access interference in the OR-MAC. In this work we show that this concatenated construction can achieve a vanishing per-user error probability in the limit of large blocklength and a large number of active users at sum-rates up to the symmetric Shannon capacity, i.e. as long as K<inf>a</inf>R &#x003C; 0.5 log<inf>2</inf>(1 + K<inf>a</inf>SNR). This extends previous point-to-point optimality results about SPARCs to the unsourced multiuser scenario. Additionally, we calculate the algorithmic threshold, that is a bound on the sum-rate up to which the inner decoding can be done reliably with the low-complexity AMP algorithm.

Unsourced Multiuser Sparse Regression Codes achieve the Symmetric MAC Capacity