An Extension of Craig's Family of Lattices

Flores, André Luiz ; Interlando, J. Carmelo ; Neto, Trajano Pires da Nóbrega

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 645-653

Résumé

Let $p$ be a prime, and let ${{\zeta }_{p}}$ be a primitive $p$ -th root of unity. The lattices in Craig's family are $(p\,-\,1)$ -dimensional and are geometrical representations of the integral $\mathbb{Z}[{{\zeta }_{p}}]$ -ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}$ , where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p\,-\,1$ where $149\,\le \,p\,\le \,3001$ , Craig's lattices are the densest packings known. Motivated by this, we construct $(p\,-\,1)(q\,-\,1)$ -dimensional lattices from the integral $\mathbb{Z}[{{\zeta }_{pq}}]$ -ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}{{\left\langle 1\,-\,{{\zeta }_{q}} \right\rangle }^{j}}$ , where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

Détail

DOI : 10.4153/CMB-2011-038-7

Mots-clés : 11H31, 11H55, 11H50, 11R18, 11R04, geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields