Additive cyclic codes over mixed alphabet $\mathbb{Z}_2\mathbb{Z}_2[u^2]\mathbb{Z}_2[u^4]$

Document Type : Original Article

Authors

Nadeem Ur Rehman ¹

Mohammad Fareed Ahmad ²

Mohd Azmi ³

¹ Department of Mathematics, Aligarh Muslim University, India

² Department of Applied Sciences and Humanities‎, ‎Parul University‎, ‎Vadodara-391760‎, ‎India

³ Department of Applied Sciences and Humanities‎, ‎Vignan Foundation for Science‎, ‎Technology and Research‎, ‎Guntur-522213‎, ‎India.

10.22034/kjm.2025.513329.3488

Abstract

Recently, there has been increasing interest in families of mixed alphabet codes that extend the scope of conventional coding theory. In this work, we study mixed alphabet codes over the structure $\mathbb{Z}_2 \mathbb{Z}_2[u^2] \mathbb{Z}_2[u^4]$, where $\mathbb{Z}_2[u^4] = \{ a_0 + u a_1 + u^2 a_2 + u^3 a_3 \mid a_i \in \mathbb{Z}_2,\, u^4 = 0 \}$. We focus on $\mathbb{Z}_2 \mathbb{Z}_2[u^2] \mathbb{Z}_2[u^4]$-additive cyclic codes, first introducing their definition and exploring their algebraic properties. We then derive the standard form of their generator matrices, along with the corresponding generator polynomials and minimal generating sets. Furthermore, we present a detailed characterization of the duals of these codes. The results contribute to the growing body of work on mixed alphabet codes, offering insights into their structural features and potential applications in error correction and related areas of information theory.\\

Keywords

Codes over ring

mixed alphabet

generator Polynomial

cyclic codes

Subjects

• 94 Information and communication, circuits

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