  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 by the
combination of two time slots or frequencies, four-dimensional
(quaternion-valued) approaches become more and more important. Hence, to
account for this fact, well-known lattice algorithms and related concepts are
generalized in this work. To this end, a brief review of complex arithmetic,
including the sets of Gaussian and Eisenstein integers, and an introduction to
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 by performing a special variant of the
Euclidean algorithm defined for matrices, 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 approaches outperform their real-valued counterparts in complexity
and/or quality aspects. Moreover, the application of the generalized algorithms
to MIMO communications is studied, particularly in the field of
lattice-reduction-aided and integer-forcing equalization.
