Quaternion SVD-SCMA for 6G Uplink: Exploiting Cross-Polarization Diversity in Hypercomplex 4D Spaces

Sixth-generation (6G) networks require advanced non-orthogonal multiple access (NOMA) schemes to support the technical requirements of dense massive machine-type communications (mMTC). While sparse-code multiple access (SCMA) improves uplink spectral efficiency, its operation within the complex two-dimensional domain (C) generates spatial congestion and high error rates under high-load scenarios. Furthermore, when using dual-polarization transceivers, cross-polarization discrimination leakage is not efficiently exploited because it operates in a conventional 2D environment. This work proposes a hypercomplex transmission architecture, called quaternionic SVD-SCMA (Q-SVD-SCMA), which maps SCMA codewords to a quaternionic group (Q8) in R4. The proposed scheme uses purely imaginary spatial rotators to project overlapping signals onto mutually orthogonal geometric subspaces, thus mitigating interference between users. On the receiver side, a quaternionic sphere decoder (Q-SD) is proposed to evaluate the minimum quaternionic Euclidean distance (MQED) to provide near-optimal detection. Computational simulations are performed under a doubly polarized Rayleigh fading channel with energy normalization to decouple arbitrary power-scale topological gains. To evaluate system performance, both the symbol error rate (SER) and the bit error rate (BER) are used. The results obtained demonstrate that Q-SVD-SCMA effectively transforms Cross-Polarization Discrimination (XPD) leakage into spatial diversity gain. The hypercomplex 4D architecture proposed in this work eliminates the interference-induced error threshold and limits the bit error penalty through multidimensional Gray mapping, providing a scalable and highly reliable physical layer framework for overloaded 6G scenarios, unlike its conventional 2D predecessors (C-SCMA and SVD-SCMA).