Improvement on asymptotic density of packing families derived from multiplicative lattices

Let ω=(−1+−3)/2. For any lattice P⊆ZnP⊆Zn, P=P+ωPP=P+ωP is a subgroup of OKn, where OK=Z[ω]⊆COK=Z[ω]⊆C. As CC is naturally isomorphic to R2R2, PP can be regarded as a lattice in R2nR2n. Let P   be a multiplicative lattice (principal lattice or congruence lattice) introduced by Rosenbloom and Tsfasman. We concatenate a family of special codes with tPℓ⋅(P+ωP), where tPtP is the generator of a prime ideal PP of OKOK. Applying this concatenation to a family of principal lattices, we obtain a new family with asymptotic density exponent λ⩾−1.26532182283λ⩾−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⩾−1.26532181404λ⩾−1.26532181404, which is better than −1.39 by considering only congruence lattice families.