Self-Dual Codes over Z_2xZ_4

TL;DR

This paper studies self-dual codes over the mixed group Z_2×Z_4, classifying their parameters, constructing examples, and analyzing their weight enumerators using invariant theory.

Contribution

It introduces three types of self-dual codes over Z_2×Z_4, establishes existence conditions, provides constructions, and describes their weight enumerators.

Findings

Classified possible parameter pairs (α,β) for each code type.

Constructed explicit examples of each code type.

Described weight enumerators using invariant theory techniques.

Abstract

Self-dual codes over $\Z_2\times\Z_4$ are subgroups of $\Z_2^\alpha \times\Z_4^\beta$ 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 $\C\subseteq \Z_2^\alpha \times\Z_4^\beta$ 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.