Even Unimodular Lattices from Quaternion Algebras

We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the E8 lattice (resp. E 2 8 and Λ16) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for constructing even unimodular lattices in any dimension multiple of 8.

[1]  Frédérique E. Oggier,et al.  Cyclic Division Algebras: A Tool for Space-Time Coding , 2007, Found. Trends Commun. Inf. Theory.

[2]  Camilla Hollanti,et al.  Well-Rounded Lattices: Towards Optimal Coset Codes for Gaussian and Fading Wiretap Channels , 2016, IEEE Transactions on Information Theory.

[3]  Gabriele Nebe,et al.  On the Euclidean minimum of some real number fields , 2005 .

[4]  Jean-Pierre Serre.  A Course in Arithmetic , 1973 .

[5]  I. Fesenko,et al.  Local fields and their extensions : a constructive approach , 1993 .

[6]  G. Nebe.  Automorphisms of extremal unimodular lattices in dimension 72 , 2014, 1409.8473.

[7]  I. Gaál.  Diophantine Equations and Power Integral Bases , 2019 .

[8]  Frédérique E. Oggier,et al.  An Error Probability Approach to MIMO Wiretap Channels , 2011, IEEE Transactions on Communications.

[9]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[10]  Pilar Bayer,et al.  Quaternion Orders, Quadratic Forms, and Shimura Curves , 2004 .

[11]  W. Fischer,et al.  Sphere Packings, Lattices and Groups , 1990 .

[12]  Andrea Baronchelli,et al.  Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer , 2019, ArXiv.

[13]  M. Gras.  Non monogénéité de l'anneau des entiers des extensions cycliques de Q de degré premier l ≥ 5 , 1986 .

[14]  Camilla Hollanti,et al.  On the Densest MIMO Lattices From Cyclic Division Algebras , 2007, IEEE Transactions on Information Theory.

[15]  Fang-Ting Tu,et al.  Lattice Packing from Quaternion Algebras , 2012 .

[16]  Julia F. Knight,et al.  Algebraic number fields , 2006 .

[17]  Frédérique E. Oggier,et al.  Algebraic Number Theory and Code Design for Rayleigh Fading Channels , 2004, Found. Trends Commun. Inf. Theory.

[18]  André Weil,et al.  Number Theory: An approach through history From Hammurapi to Legendre , 1984 .

[19]  B. A. Sethuraman.  Division Algebras and Wireless Communication , 2009, ArXiv.

[20]  Frédérique E. Oggier,et al.  Lattice Code Design for the Rayleigh Fading Wiretap Channel , 2011, 2011 IEEE International Conference on Communications Workshops (ICC).

[21]  S. Costa,et al.  Algebraic construction of lattices via maximal quaternion orders , 2020 .

[22]  E. Bayer-Fluckiger.  Lattices and number Fields , 1999 .

[23]  Johanna Weiss,et al.  Arithmetique Des Algebres De Quaternions , 2016 .

[24]  I. Gaál,et al.  Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields , 2014, 1810.00060.

[25]  Jürgen Klüners,et al.  On the negative Pell equation , 2010 .

[26]  J. Martinet,et al.  Formes quadratiques liées aux algèbres semi-simples. , 1994 .

[27]  Theodor Estermann.  Einige Sätze über quadratfreie Zahlen , 1931 .