Algebraic constructions of rotated unimodular lattices and direct sum of Barnes-Wall lattices

Abstract

In this paper, we construct some families of rotated unimodular lattices and rotated direct sum of Barnes–Wall lattices BWn for n=4,8 and 16 via ideals of the ring of the integers Z[ζ2rq+ζ−12rq] for q=3,5 and 15. We also construct rotated BW16 and BW32-lattices via Z-submodules of Z[ζ2r15+ζ−12r15]. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here.