Abstract
Well-rounded lattices have been considered in coding theory, in approaches to MIMO, and SISO wiretap channels. Algebraic lattices have been used to obtain dense lattices and in applications to Rayleigh fading channels. Recent works study the relation between well-rounded lattices and algebraic lattices, mainly in dimension two. In this article we present a construction of well-rounded algebraic lattices in Euclidean spaces of odd prime dimension. We prove that for each Abelian number field of odd prime degree having squarefree conductor, there exists a \({\mathbb {Z}}\)-module M such that the canonical embedding applied to M produces a well-rounded lattice. It is also shown that for each odd prime dimension there are infinitely many non-equivalent well-rounded algebraic lattices, with high indexes as sublattices of other algebraic lattices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bayer-Fluckiger, E., Oggier, F., Viterbo, E.: New algebraic constructions of rotated \( {Z}^n\)-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inf. Theory 50(4), 702–714 (2004)
Boutros, J., Viterbo, E., Rastello, C., Belfiore, J.: Good lattice constellations for both Rayleigh fading and Gaussian channel. IEEE Trans. Inf. Theory 42(2), 502–517 (1996)
Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Group, 3rd edn. Springer, Nova Iorque (1998)
Fukshansky, L., Petersen, K.: On well-rounded ideal lattices. Int. J. Number Theory 8(1), 189–206 (2012)
Gnilke, O.W., Tran, H.T.N., Karilla, A., Hollanti, C.: Well-rounded lattices for reliability and security in Rayleigh fading SISO Channels. In: IEEE Information Theory Workshop (ITW), pp. 359–363 (2016)
Gnilke, O.W., Barreal, A., Karrila, A., Tran, H.T.N., Karpuk, D., Hollanti, C.: Well-rounded lattices for coset coding in MIMO wiretap channels. IEEE International Telecommunication Newtworks and Applications Conference, Dunedin, New Zeland (2016)
Jorge, G.C., Costa, S.I.R.: On rotated \(D_n\)-lattices constructed via totally real number fields. Arch. Math. 100, 323–332 (2013)
Martinet, J.: Perfect Lattices in Euclidean Spaces. Springer, Berlin (2003)
McMullen, C.T.: Minkowski’s conjecture, well-rounded lattices and topological dimension. J. Am. Math. Soc. 18(3), 711–735 (2005)
Nunes, J.V.L., Interlando, J.C., Neto, T.P.N., Lopes, J.O.D.: New \(p\)-dimensional lattices from cyclic extensions. J. Algebra Appl. 16, 175–186 (2017)
Oliveira, E.L., Interlando, J.C., Neto, T.P.N., Lopes, J.O.D.: The integral trace form of cyclic extensions of odd prime degree. Rocky Mountain J. Math 47(4), 1075–1088 (2017)
Samuel, P.: Algebraic Theory of Numbers. Hermann, Paris (1970)
Washington, L.: Introduction to Cyclotomic Fields, 2nd edn. Springer, New York (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by CAPES, under Grant 1540410, FAPESP, under Grant 13/25977-7, and CNPq, under Grant 312926/2013-8.
Rights and permissions
About this article
Cite this article
de Araujo, R.R., Costa, S.I.R. Well-rounded algebraic lattices in odd prime dimension. Arch. Math. 112, 139–148 (2019). https://doi.org/10.1007/s00013-018-1232-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-018-1232-7