Algorithms and Bounds for Complex and Quaternionic Lattices with Application to MIMO Transmission

Lattices are a popular field of study in mathematical research, but also more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In theory, most often lattices over real numbers considered. However, communications, complex-valued processing is usually interest. Besides, by the use dual-polarized transmission as well combination two time slots frequencies, four-dimensional (quaternion-valued) approaches become and important. Hence, to account for this fact, well-known lattice algorithms related concepts generalized work. To end, brief review complex arithmetic, including sets Gaussian Eisenstein integers, an introduction quaternion-valued numbers, Lipschitz Hurwitz given. On that basis, variants important derived: first, polynomial-time LLL algorithm, resulting reduced basis performing special variant Euclidean algorithm defined matrices, second, calculate successive minima—the norms shortest independent vectors lattice—and its points. Generalized bounds quality particular results established asymptotic complexities assessed. These findings extensively compared conventional real-valued processing. It shown outperform their counterparts complexity and/or aspects. Moreover, application MIMO communications studied, particularly lattice-reduction-aided integer-forcing equalization.