Computing the generator polynomials of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes
نویسندگان
چکیده
A Z2Z4-additive code C ⊆ Z α 2 × Z β 4 is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z2 and the set of Z4 coordinates, such that any simultaneous cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the Z4[x]-module Z2[x]/(x − 1)×Z4[x]/(x −1). Any Z2Z4-additive cyclic code C is of the form 〈(b(x) | 0), (l(x) | f(x)h(x)+2f(x))〉 for some b(x), l(x) ∈ Z2[x]/(x −1) and f(x), h(x) ∈ Z4[x]/(x − 1). A new algorithm is presented to compute the generator polynomials for Z2Z4additive cyclic codes.
منابع مشابه
Some results of linear codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4$
Abstract: In this paper, we mainly study the theory of linear codes over the ring R = Z4 + uZ4 + vZ4 + uvZ4. By the Chinese Remainder Theorem, we have R is isomorphic to the direct sum of four rings Z4. We define a Gray map Φ from R n to Z 4 , which is a distance preserving map. The Gray image of a cyclic code over R is a linear code over Z4. Furthermore, we study the MacWilliams identities of ...
متن کامل$(1+2u)$-constacyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$
Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic c...
متن کاملOn $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-$(1+u)$-additive constacyclic
In this paper, we study Z2Z2[u]-(1 + u)-additive constacyclic code of arbitrary length. Firstly, we study the algebraic structure of this family of codes and a set of generator polynomials for this family as a (Z2+uZ2)[x]-submodule of the ring Rα,β. Secondly, we give the minimal generating sets of this family codes, and we determine the relationship of generators between the Z2Z2[u]-(1 + u)-add...
متن کاملRelative two-weight $\mathbb{Z}_2 \mathbb{Z}_4$-additive Codes
In this paper, we study a relative two-weight Z2Z4-additive codes. It is shown that the Gray image of a two-distance Z2Z4-additive code is a binary two-distance code and that the Gray image of a relative two-weight Z2Z4-additive code, with nontrivial binary part, is a linear binary relative two-weight code. The structure of relative two-weight Z2Z4-additive codes are described. Finally, we disc...
متن کاملCyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$
In this paper, we have studied cyclic codes over the ring R = Z4 +uZ4, u = 0. We have considered cyclic codes of odd lengths. A sufficient condition for a cyclic code over R to be a Z4-free module is presented. We have provided the general form of the generators of a cyclic code over R and determined a formula for the ranks of such codes. In this paper we have mainly focused on principally gene...
متن کامل