An Extension of Craig's Family of Lattices

Abstract

Abstract. Let p be a prime, and let ζp 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[ζp]-ideals 1 − ζp 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[ζpq]-ideals 1 − ζp i 1 − ζq 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.