An Extension of Craig's Family of Lattices

Abstract

AbstractLet 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.