I. Introduction
Overloaded systems, in which the number of users is greater than the dimension of the signal-space, are of practical interest in bandwidth-efficient multiuser communication. One kind of such systems is sparse code multiple access (SCMA), which is a promising code/frequency domain non-orthogonal multiple access technique to address the challenges for the fifth generation (5G) mobile networks [1], [2]. Applications of non-orthogonal signaling for multiuser communications have been investigated long ago, significant efforts were paid to the optimal signaling design and intensive multiuser detection techniques, to suppress the multiple access interference (MAI) for lowering probability of error or increasing capacity. For example, the spreading signatures invoking hierarchy of orthogonal subsets and the classes of uniquely detectable sequences were proposed to handle the overloaded code-division multiple access (CDMA) [3]–[5]. Also, sophisticated multiuser detection schemes, such as the optimum joint detector, the suboptimal linear minimum mean-square error (MMSE) detector, and the iterative serial interference cancelation, were developed to mitigate the MAI to various degrees [6]–[10]. Moreover, signature of low density type, or sparsely spread CDMA, was suggested, to reduce both the MAI and the complexity of optimum multiuser detection [11]–[13]. Such kind of low density signature (LDS) enables the near-optimal multiuser detection using message-passing algorithm (MPA) with low-complexity. Inspired by the overloading capability and the low-complexity feature of LDS, SCMA was developed by inheriting from LDS the sparse sequences structure, such that MPA is available for achieving near-optimal performance. In contrast to the LDS scheme, multidimensional signal constellations, instead of the repetition coding, were utilized in SCMA to combat the channel fading. As a result, the larger coding gain and better spectrum efficiency were achievable for SCMA due to the improved constellations/codebooks, compared to LDS.