Hypercomplex Codebooks Optimized by MQED and Low-Complexity Spherical Detection

6G uplink networks demand high spectral efficiency for massive machine-type communications (mMTCs), where overloading sparse code multiple access (SCMA) suffers from inter-user interference (IUI) in the two-dimensional complex plane and wastes the energy filtered by cross-polarization discrimination (XPD) of dual-polarized antennas. Regarding the hypercomplex Q-SVD-SCMA architecture, this work contributes three advances. First, it identifies that the unoptimized Q8 alphabet exhibits a zero minimum quaternionic Euclidean distance (MQED) at the superposition level and proposes an optimized quaternionic codebook that maximizes the MQED, increasing it from 0 to 0.447 and reducing the symbol error rate (SER) from approximately 33 × to 24 dB compared to Q8 in the diagonalized channel regime characteristic of the Q-SVD-SCMA architecture, also outperforming an optimized complex codebook. Second, an exact quaternionic-sphere decoder (Q-SD) is presented that replicates maximum likelihood (ML) performance by visiting an average of approximately dc⋅M candidates, an empirical reduction of approximately 21 × compared to exhaustive search, with decreasing complexity as SNR increases. Third, robustness to imperfect channel state (CSI) is evaluated using a coherent Gauss–Markov model, showing graceful degradation and an order-of-magnitude advantage over 2D baselines. The results position hypercomplex SCMA as a robust, low-complexity candidate for the 6G uplink.