Improvement on asymptotic density of packing families derived from multiplicative lattices

Let ω = (−1 + √ −3)/2. For any lattice P ⊆ Zn, P = P + ωP is a subgroup of On K , where OK = Z[ω] ⊆ C. As C is naturally isomorphic to R2, P can be regarded as a lattice in R2n. Let P be a multiplicative lattice (principal lattice or congruence lattice) introduced by Rosenbloom and Tsfasman. We concatenate a family of special codes with tP · (P + ωP ), where tP is the generator of a prime ideal P of OK . Applying this concatenation to a family of principal lattices, we obtain a new family with asymptotic density exponent λ > −1.26532182283, which is better than −1.87 given by Rosenbloom and Tsfasman considering only principal lattice families. For a new family based on congruence lattices, the result is λ > −1.26532181404, which is better than −1.39 by considering only congruence lattice families.