An Extension of Craig's Family of Lattices
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Data
2011-12-01
Autores
Flores, Andre Luiz
Interlando, J. Carmelo
da Nobrega Neto, Trajano Pires [UNESP]
Título da Revista
ISSN da Revista
Título de Volume
Editor
Canadian Mathematical Soc
Resumo
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 Z[zeta(p)]-ideals < 1 - zeta(p)>(i), where i is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions p - 1 where 149 <= p <= 3001, Craig's lattices are the densest packings known. Motivated by this, we construct (p - 1)(q - 1)-dimensional lattices from the integral Z[zeta(pq)]-ideals < 1 - zeta(p)>(i) < 1 - zeta(q)>(j), where p and q are distinct primes and i and fare 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.
Descrição
Palavras-chave
geometry of numbers, lattice packing, Craig's lattices, Quadratic form, Cyclotomic fields
Como citar
Canadian Mathematical Bulletin-bulletin Canadien de Mathematiques. Ottawa: Canadian Mathematical Soc, v. 54, n. 4, p. 645-653, 2011.