Codes for the Lee metric and lattices for the l1-distance

Forney has proposed an iterated construction called the squaring construction for simplified derivation and representation of the Barnes-Wall lattices. He used as a starting partition chain the two-dimensional infinite two-way partition...Z²/ Z²/ RZ²/ 2Z²/ 2RZ²...with minimum Euclidean distances 1/1/1/2/4/8/..., where R is a two-dimensional rotation operator. We apply this construction to the one-dimensional infinite two-way partition...Z/Z/2Z/4Z/8Z...with minimum distance...1/1/2/4/8/...which has clearly the same properties as the previous partition. The resulting lattices of dimension N=2ⁿ for the l₁-distance can therefore be regarded as the duals of the Barnes-Wall lattices of dimension 2N for the Euclidean distance. Since the 2-depth of each of these lattices is equal to n they necessarily contain the 2ⁿZ^N lattice. The coset representatives of these lattices in ν2ⁿZ^N, where ν is an arbitrary nonnegative integer, are good codes for the Lee distance since they outperform the negacyclic codes in low dimensions. Maximum likelihood (ML) soft detection can be performed easily on these lattices and codes since they have a simple trellis structure