Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$

$$\mathbb{Z }_2\mathbb{Z }_4$$ -Additive formally self-dual codes

“…Therefore, the code is an odd separable formally self-dual code. Also, unlike the case for self-dual codes [5], separability does not imply that the code is antipodal. For example, in the separable code given above, (10|) × |2 , the code is separable but not antipodal; i.e., (11|2) is not in the code.…”

“…For a Z 2 Z 4 -additive self-dual code C, there are some conditions that relates separability, antipodality and the Type of the code as it was proved in [5]. If C is an antipodal code, then C is of Type I or Type II.…”

“…Following the terminology of [5] and [17], where the definitions were made for self-dual codes, we define the following terms.…”

“…In [10], and earlier in [31], the shadow of binary codes was considered. Namely, given a Type I binary self-dual code, the weight enumerator of the subcode C of doubly-even vectors is computed by the formula in (5). Then, the weight enumerator of the shadow C ⊥ − C is computed using the MacWilliams relations.…”

“…Building up constructions were first given for binary codes in [9]. They were extended in many places, for example in [15], [18], [19], and [5]. Recently, these ideas were extended to formally self-dual codes in [24] and [16].…”