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

Abstract

Self-dual codes over $\mathbb Z_2\times\mathbb Z_4$ are subgroups of $\mathbb Z_2^\alpha\times\mathbb Z_4^\beta$ that are equal to their orthogonal under an inner-product that relates these codes to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $\alpha,\beta$ such that there exist a self-dual code $\mathcal C\subseteq \mathbb Z_2^\alpha \times\mathbb Z_4^\beta$ are established. Moreover, the construction of such a code for each type and possible pair $(\alpha,\beta)$ is given. The standard techniques of invariant theory are applied to describe the weight enumerators for each type. Finally, we give a construction of self-dual codes from existing self-dual codes.