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

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.