Double cyclic and Quantum codes

Abstract

In this paper, we explore the structure of \( \mathbb{Z}_2 + u \mathbb{Z}_2 = \{0, 1, u, u + 1\} = \mathbb{Z}_2[u]\)-additive codes, where $u^2=0$ and their generalization to double cyclic codes. We establish the algebraic framework for these codes over the ring \( \mathbb{Z}_2[u] \) and its extensions. Additionally, we provide explicit generators for double cyclic codes and define the Gray map to derive corresponding binary linear codes and quantum codes. Finally, we present an example illustrating the construction of a binary linear code.