Near-ellipsoidal Voronoi coding

We consider a special case of Voronoi coding, where a lattice Λ in Rⁿ is shaped (or truncated) using a lattice Λ´={(m₁x₁,...,m_nx_n)|(x₁,...,x_n)∈Λ} for a fixed m_=(m₁,...,m_n)∈(N/{0,1})ⁿ. Using this technique, the shaping boundary is near-ellipsoidal. It is shown that the resulting codes can be indexed by standard Voronoi indexing algorithms plus a conditional modification step, as far as Λ´ is a sublattice of Λ. We derive the underlying conditions on m_ and present generic near-ellipsoidal Voronoi indexing algorithms. Examples of constraints on m_ and conditional modification are provided for the lattices A₂, D_n (n≥2) and 2D_n⁺ (n even ≥4).