On the construction of some type II codes over ℤ4 × ℤ4

We consider here the construction of Type II codes over the abelian group ℤ4 × ℤ4. The definition of Type II codes here is based on the definitions introduced by Bannai [2]. The emphasis is given on the construction of these types of codes over the abelian group ℤ4 × ℤ4 and in particular, the methods applied by Gaborit [7] in the construction of codes over ℤ4 was extended to four different dualities with their corresponding weight functions (maps assigning weights to the alphabets of the code). In order to do this, we use the flattened form of the codes and construct binary codes analogous to the ones applied to ℤ4 codes. Since each duality generates more than one weight function, we focus on those weights satisfying the squareness property. Here, by the squareness property, we mean that the weight function wt assigns the weight 0 to the ℤ4 × ℤ4 elements (0, 0), (2, 2) and the weight 4 to the elements (0, 2) and (2, 0). The main results of this paper are focused on the characterization of these codes and provide a method of construction that can be applied in the generation of such codes whose weight functions satisfy the squareness property