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( Lattice packing and covering [See also 05B40, 52C15, 52C17] )
1. CMB 2011 (vol 54 pp. 645)
 Flores, André Luiz; Interlando, J. Carmelo; Neto, Trajano Pires da Nóbrega

An Extension of Craig's Family of Lattices
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.
Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields Categories:11H31, 11H55, 11H50, 11R18, 11R04 
