Understanding the fundamental limits of technologies enabling future wireless communication systems is essential for their efficient state-of-the-art design. A prominent technology of major interest in this framework is non-orthogonal multiple access (NOMA). In this paper, we derive an explicit rigorous closed-form analytical expression for the optimum spectral efficiency, in the large-system limit, of regular sparse (code-domain) NOMA, along with a closed-form formulation for the limiting spectral density of the underlying input covariance matrix. Sparse NOMA is an attractive and popular instantiation of NOMA, and its ‘regular’ variant corresponds to the case where only a fixed and finite number of orthogonal multiple-access resources are allocated to any designated user, and vice versa. Interestingly, the well-known Verdú-Shamai formula for (dense) randomly-spread code-division multiple access (RS-CDMA) turns out to coincide with the limit of the derived expression, when the number of orthogonal resources per user grows large. Furthermore, regular sparse NOMA is rigorously shown to be spectrally more efficient than RS-CDMA (viz. dense NOMA) across the entire system load range. Therefore, regular sparse NOMA may serve as an appealing means for reducing the throughput gap to orthogonal transmission in the underloaded regime, and to the ultimate Cover-Wyner bound for massive access overloaded systems. The spectral efficiency is also derived in closed form for the sub-optimal linear minimum-mean-square-error (LMMSE) receiver, which again extends the corresponding Verdú-Shamai LMMSE formula to the case of regular sparse NOMA.
