Correction: Linearity of $$\mathbb {Z}_{2^L}$$-linear codes via Schur product

Squares of matrix-product codes

Most practical constructions of lattice codes with high coding gains are multilevel constructions where each level corresponds to an underlying code component. Construction D, Construction $$\hbox {D}'$$ D ? , and Forney's code formula are classical constructions that produce such lattices explicitly from a family of nested binary linear codes. In this paper, we investigate these three closely related constructions along with the recently developed Construction $$\hbox {A}'$$ A ? of lattices from codes over the polynomial ring $$\mathbb {F}_2[u]/u^a$$ F 2 [ u ] / u a . We show that Construction by Code Formula produces a lattice packing if and only if the nested codes being used are closed under Schur product, thus proving the similarity of Construction D and Construction by Code Formula when applied to Reed---Muller codes. In addition, we relate Construction by Code Formula to Construction $$\hbox {A}'$$ A ? by finding a correspondence between nested binary codes and codes over $$\mathbb {F}_2[u]/u^a$$ F 2 [ u ] / u a . This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction $$\hbox {A}'$$ A ? . Finally, we show that Construction $$\hbox {A}'$$ A ? produces a lattice if and only if the corresponding code over $$\mathbb {F}_2[u]/u^a$$ F 2 [ u ] / u a is closed under shifted Schur product.

A characterization of MDS codes that have an error correcting pair