On the construction of some Type II codes over \({\mathbb Z}_4{\times}{\mathbb Z}_4\) (Q1412257)

The author describes four different dualities for codes over the Abelian group \(\mathbb Z_4 \times \mathbb Z_4.\) For each duality, weight functions satisfying the squareness property are given for the elements of the group. With a given duality and weight function Type~II codes are defined as codes satisfying \(C=C^\perp\), where \(C^\perp\) is the orthogonal under the given duality, and \(2 \ell \) divides \(\text{wt}(v)\) for all \(v \in C,\) where \(\ell\) is the largest order of the elements of the group. The author proves construction theorems for these Type~II codes and theorems for the structure of these codes similar to those for codes over \(\mathbb Z_4.\)