Self-Dual Codes over Z_2xZ_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 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 code $\C\subseteq \Z_2^α\times\Z_4^β$ are established. Moreover, the construction of a $\add$-linear code for each type and possible pair $(\alpha ,\beta )$ is given. Finally, the standard techniques of invariant theory are applied to describe the weight enumerators for each type.

Submitted to Designs, Codes and Cryptography