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

Abstract

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