Algebraic construction of lattices via maximal quaternion orders

Abstract

In this paper we propose a framework to construct algebraic lattices in dimensions 4n via ideals from maximal orders of a quaternion algebra whose center is a totally real number field. For n=1,2,3,4 and 6 it was possible to construct rotated versions of the densest lattices in their dimensions, D4,E8,K12,Λ16 and Λ24. We also present a family of lattices in dimension 2r from A=(−1,−1)Q(ζ2r +ζ2r −1) and a characterization of a maximal quaternion order of A by using the Chebyshev polynomials.