$\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes

TL;DR

This paper introduces and analyzes a new class of additive cyclic codes over a product of rings, exploring their structure, duals, and connections to binary codes through Gray maps.

Contribution

It defines $ ext{Z}_{2^i}$-additive cyclic codes, establishes their algebraic structure, duality properties, and provides a polynomial framework for their analysis.

Findings

Codes are characterized as submodules of polynomial rings.

Dual codes are also cyclic, preserving structure.

An example illustrates the code construction.

Abstract

In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$ and $\rm{i}$ being relatively prime for each $i=1,2,\ldots,n.$ We first define a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then define the distance between two codewords and the minimum distance of such a code. Moreover we relate these to binary codes using the generalized Gray maps. We define the duals of such codes and show that the dual of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code is also cyclic. We then give the polynomial definition of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code of a certain length. We then determine the structure of such codes and derive a minimal spanning set for that. We also determine the total number of codewords in this code. We finally give an illustrative example of a $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}$-additive cyclic code.