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# An Extension of Craig's Family of Lattices

Published:2011-03-10
Printed: Dec 2011
• André Luiz Flores,
Departamento de Matemática, Universidade Federal de Alagoas, Arapiraca, AL, Brazil
• J. Carmelo Interlando,
Department of Mathematics and Statistics, San Diego State University, San Diego, CA, U.S.A.
• Trajano Pires da Nóbrega Neto,
Departamento de Matemática, Universidade Estadual Paulista, São José do Rio Preto, SP, Brazil
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## Abstract

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 $\langle 1-\zeta_p \rangle^i$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$, 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 $\langle 1-\zeta_p \rangle^i \langle 1-\zeta_q \rangle^j$, 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.
 Keywords: geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields
 MSC Classifications: 11H31 - Lattice packing and covering [See also 05B40, 52C15, 52C17] 11H55 - Quadratic forms (reduction theory, extreme forms, etc.) 11H50 - Minima of forms 11R18 - Cyclotomic extensions 11R04 - Algebraic numbers; rings of algebraic integers

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