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

In this paper we study n ∏ i=1 Z2i -Additive Cyclic Codes. These codes are identified as Z2n [x]submodules of n ∏ i=1 Z2i [x]/〈x αi − 1〉; αi and i being relatively prime for each i = 1, 2, . . . , n. We first define a n ∏ i=1 Z2i -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 n ∏ i=1 Z2i -additive cyclic code is also cyclic. We then give the polynomial definition of a n ∏ i=1 Z2i 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 n ∏ i=1 Z2i -additive cyclic code. Index Terms Additive code, Cyclic code, Dual Code, Ideal, Generator, Polynomial, Spanning set