Abstract view
An Extension of Craig's Family of Lattices


Published:20110310
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
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 $(p1)$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 $p1$ where $149 \leq p \leq 3001$,
Craig's lattices are the densest packings known. Motivated by this,
we construct $(p1)(q1)$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 spherepacking 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 spherepacking properties.
MSC Classifications: 
11H31, 11H55, 11H50, 11R18, 11R04 show english descriptions
Lattice packing and covering [See also 05B40, 52C15, 52C17] Quadratic forms (reduction theory, extreme forms, etc.) Minima of forms Cyclotomic extensions Algebraic numbers; rings of algebraic integers
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
