Lattice-reduction-aided preequalization or precoding are powerful techniques for handling the interference on the multi-user MIMO broadcast channel as the channel's diversity order can be achieved. However, recent advantages in the closely related field of integer-forcing equalization raise the question, if the unimodularity constraint on the integer equalization matrix in LRA schemes is really necessary or if it can be dropped, yielding an additional factorization gain. In this paper, socalled algebraic signal constellations are presented, where the unimodularity is not required anymore. Assuming complexbaseband transmission, particularly q-ary fields of Gaussian primes (complex integer lattice) and Eisenstein primes (complex hexagonal lattice) are considered. Given the signal constellation and the channel code in the same arithmetic over a finite field of order q, a coded modulation approach with straightforward softdecision decoding metric is applied. Moreover, LRA precoding over algebraic constellations and its advantages as opposed to LRA preequalization are discussed. The theoretical considerations in the paper are covered by means of numerical simulations.
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