Quaternionic Minkowski Reduction and Its Application in Dual-Polarized MIMO Systems

The task of finding the shortest vectors of a lattice that form a basis thereof is known as Minkowski reduction. In this work, Minkowski reduction is adapted to discrete subgroups of quaternionic spaces defined over the set of Hurwitz integers which constitutes a Euclidean subring of the (Hamilton) quaternions. Based on a quaternionic variant of Minkowski's greatest-common-divisor condition for suitable lattice vectors, a lattice-basis-reduction algorithm is derived. To that end, a basisupdate approach is presented which takes the non-commutativity of quaternionic multiplication into account. The reduction algorithm is finally applied in multi-user multiple-input/multipleoutput (MIMO) systems with dual-polarized antennas, where the quaternionic interference is handled by lattice-reduction-aided equalization. It is shown that quaternionic Minkowski reduction achieves significantly better results than state-of-the-art Lenstra-Lenstra-Lovász (LLL) reduction and that the successive minima of quaternionic lattices are approximated quite well.