Improvement on Asymptotic Density of Packing Families Derived from Multiplicative Lattices

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Let ω=(-1+√-3)/2. For any lattice P Zⁿ, P=P+ωP is a subgroup of Oₖⁿ, 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 tₚℓ·(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.