Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
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A Gilbert-Varshamov-type bound for Euclidean packings was recently found by Nebe and Xing. In this present paper, we derive a Gilbert-Varshamov-type bound for lattice packings by generalizing Rush's approach of combining p-ary codes with the lattice pZ^n. Specifically, we will exploit suitable sublattices of Z^n as well as lattices of number fields in our construction. Our approach allows us to compute the center densities of lattices of moderately large dimensions which compare favorably with the best known densities given in the literature as well as the densities derived directly via Rush's method.