Algebraic construction of lattices via maximal quaternion orders

摘要

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, D-4, E-8, E-12, A(16) and A(24). We also present a family of lattices in dimension 2(r) from A = (-1, 1)(Q(zeta 2r + zeta 2r-1)) and a characterization of a maximal quaternion order of A by using the Chebyshev polynomials. (C) 2019 Elsevier B.V. All rights reserved.