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Sparse Code Multiple Access (SCMA) has received much attention in recent years due to its ability to provide higher spectral efficiency, a requirement of 5G future wireless communications, and beyond. SCMA is a code domain Non-Orthogonal Multiple Access (NOMA) scheme, and is developed from the class of Low-Density Signature (LDS) structure of Code Division Multiple Access (CDMA). It has many possible uses in device-to-device (D2D) communications (Sultana et al., 2020) and has also been investigated as a potential candidate for 5G (Chen, et al., 2016). Research reveals that SCMA achieves higher spectral efficiency when lattice theory is adopted in the design of its mother constellation (Zhang et al., 2019).
One way to design the mother constellation of SCMA is a complex lattice constellation. There has been extensive research regarding the constellation design based on a rotated lattice to improve the system reliability (Boutros & Viterbo, 1998). For example, Nikopour and Baligh (2013) showed that the SCMA mother constellation, which is also known as the SCMA encoder, could be defined as a map of the user information into sparse codewords of the mother constellation. Moreover, Mahmoud et al. (2014) explored the construction of a mother constellation by utilizing two small-size QAM constellations and applying a Cartesian product on these constellations to produce SCMA constellation points. Furthermore, lattice rotation is defined to realize a unique codebook for each user. Lisu et al. (2015) employed a star QAM to formulate an SCMA mother constellation; they compared their system performance with that in Taherzadeh et al. (2014) and the LDS system performance over the Rayleigh fading channel. Bao et al. (2016) introduced a multiuser codebook design of SCMA based on cutoff rate analysis of multi-input multi-output (MIMO) systems. Cai et al. (2016) designed a multidimensional codebook for downlink SCMA transmission; they stated that the multidimensional codebooks outperform the performance of LDS (Beek & Popovic, 2009) and the codebook design in Taherzadeh et al. (2014). More recently, Ma et al. (2019) focused on modeling the SCMA codebook for Satellite Internet of Things (IoT) to reduce the Peak-to-Average Power Ratio (PAPR). However, although the previous works have focused only on rotating the complex lattice in constructing the SCMA codebook, little attention has been paid to modeling the SCMA codebook based on an algebraic real number field and ideal lattice 
for improving the system reliability.
In the present study, we implement the mother constellation of SCMA based on the cyclotomic construction and a totally real number field. The SCMA mother constellation thus designed generally ensured that the diversity of any adjacent vectors (points) in the mother constellation equals the length of those vectors (points). This property increases the minimum product distance between any adjacent points in the mother constellation, and, consequently, decreases the probability of decoding error. This combination of SCMA and the cyclotomic construction forms a novel mother constellation in which the codewords are real vectors. What is more, designing a real mother constellation (codebook) for the SCMA provides a reduced probability of decoding error, also known as the Bit Error Rate (BER), performance over the complex codebook of SCMA.