Z-Opt: A Near-Optimal Reduced-Complexity Two-Dimensional Grassmannian Constellation

Grassmannian constellations are known to achieve the capacity of noncoherent communications over Rayleigh fading channels in the high-SNR regime, yet their efficient construction remains challenging. In this paper, we propose two construction methods for Grassmannian constellations of one-dimensional subspaces in a two-dimensional space, termed S-Opt and Z-Opt, along with two low-complexity detectors. Both the construction and detection procedures are performed on the unit sphere, known as the Bloch sphere in quantum computing. We show that the chordal distance on the Grassmann manifold is proportional to the Euclidean distance on the Bloch sphere and derive a corresponding theoretical upper bound based on the Fejes--T\'oth bound on the minimum chordal distance. The S-Opt constellation is constructed from sphere-packing solutions and attains the derived upper bound for the optimal Bloch-sphere packings considered. The S-Opt detector can be applied to arbitrary Grassmannian constellations on $\mathcal{G}(2,1)$, and its time complexity scales linearly with the number of receive antennas and logarithmically with the constellation size, while yielding the same detection performance as the GLRT detector. Furthermore, based on the insight obtained through the S-Opt construction, the Z-Opt constellation is constructed by stacking regular polygons on the Bloch sphere, and its minimum chordal distance approaches the derived upper bound over the evaluated constellation sizes. The Z-Opt detector's time complexity scales linearly with the number of receive antennas, while yielding the same detection performance as the GLRT detector for Z-Opt.