Self-Dual Codes over $\mathbb{Z}_2\times (\mathbb{Z}_2+u\mathbb{Z}_2)$

TL;DR

This paper investigates self-dual codes over a specific algebraic structure combining binary and ring components, classifies possible parameters, proposes construction methods, and characterizes two-weight codes.

Contribution

It introduces three types of self-dual codes over the algebra, determines feasible parameters, and fully characterizes two-weight codes for certain cases.

Findings

Classified possible parameter values for self-dual codes.

Developed multiple construction approaches.

Fully characterized two-weight self-dual codes.

Abstract

In this paper, we study self-dual codes over $\mathbb{Z}_2 \times (\mathbb{Z}_2+u\mathbb{Z}_2) $, where $u^2=0$. Three types of self-dual codes are defined. For each type, the possible values $\alpha,\beta$ such that there exists a code $\mathcal{C}\subseteq \mathbb{Z}_{2}^\alpha\times (\mathbb{Z}_2+u\mathbb{Z}_2)^\beta$ are established. We also present several approaches to construct self-dual codes over $\mathbb{Z}_2 \times (\mathbb{Z}_2+u\mathbb{Z}_2) $. Moreover, the structure of two-weight self-dual codes is completely obtained for $\alpha \cdot\beta\neq 0$.