Negacyclic codes over Z4+uZ4

In this paper, we study negacyclic codes of odd length and of length $2^k$ over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2=0$. We give the complete structure of negacyclic codes for both the cases. We have obtained a minimal spanning set for negacyclic codes of odd lengths over $R$. A necessary and sufficient condition for negacyclic codes of odd lengths to be free is presented. We have determined the cardinality of negacyclic codes in each case. We have obtained the structure of the duals of negacyclic codes of length $2^k$ over $R$ and also characterized self-dual negacyclic codes of length $2^k$ over $R$.Comment: 18 page

A class of cyclic Codes Over the Ring Z4[u]/<u2>\Z_4[u]/<u^2> and its Gray image

Cyclic codes over R have been introduced recently. In this paper, we study the cyclic codes over R and their $\Z_2$ image. Making use of algebraic structure, we find the some good $\Z_2$ codes of length 28.Comment: 10 page

An explicit representation and enumeration for negacyclic codes of length 2kn2^kn over Z4+uZ4\mathbb{Z}_4+u\mathbb{Z}_4

In this paper, an explicit representation and enumeration for negacyclic codes of length $2^kn$ over the local non-principal ideal ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$ $(u^2=0)$ is provided, where $k, n$ are any positive integers and $n$ is odd. As a corollary, all distinct negacyclic codes of length $2^k$ over $R$ are listed precisely. Moreover, a mass formula for the number of negacyclic codes of length $2^kn$ over $R$ is given and a mistake in [Cryptogr. Commun. (2017) 9: 241--272] is corrected

Some results of linear codes over the ring Z4+uZ4+vZ4+uvZ4\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4

In this paper, we mainly study the theory of linear codes over the ring $R =\mathbb{Z}_4+u\mathbb{Z}_4+v\mathbb{Z}_4+uv\mathbb{Z}_4$. By the Chinese Remainder Theorem, we have $R$ is isomorphic to the direct sum of four rings $\mathbb{Z}_4$. We define a Gray map $\Phi$ from $R^{n}$ to $\mathbb{Z}_4^{4n}$, which is a distance preserving map. The Gray image of a cyclic code over $R^{n}$ is a linear code over $\mathbb{Z}_4$. Furthermore, we study the MacWilliams identities of linear codes over $R$ and give the the generator polynomials of cyclic codes over $R$. Finally, we discuss some properties of MDS codes over $R$

(1+2u)(1+2u)-constacyclic codes over Z4+uZ4\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 codes of length $2n$ over $\mathbb{Z}_4$. Further, a number of linear codes over $\mathbb{Z}_4$ as the images of $(1+2u)$-constacyclic codes over $R$ are obtained

Negacyclic codes over the local ring Z4[v]/⟨v2+2v⟩\mathbb{Z}_4[v]/\langle v^2+2v\rangle of oddly even length and their Gray images

Let $R=\mathbb{Z}_{4}[v]/\langle v^2+2v\rangle=\mathbb{Z}_{4}+v\mathbb{Z}_{4}$ ($v^2=2v$) and $n$ be an odd positive integer. Then $R$ is a local non-principal ideal ring of $16$ elements and there is a $\mathbb{Z}_{4}$-linear Gray map from $R$ onto $\mathbb{Z}_{4}^2$ which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over $R$ of length $2n$ are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over $R$ of length $2n$ are presented. Moreover, all $23\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p}-2}{p}}$ negacyclic codes over $R$ of length $2M_p$ and all $3\cdot(4^p+5\cdot 2^p+9)^{\frac{2^{p-1}-1}{p}}$ self-dual codes among them are presented precisely, where $M_p=2^p-1$ is a Mersenne prime. Finally, $36$ new and good self-dual $2$-quasi-twisted linear codes over $\mathbb{Z}_4$ with basic parameters $(28,2^{28}, d_L=8,d_E=12)$ and of type $2^{14}4^7$ and basic parameters $(28,2^{28}, d_L=6,d_E=12)$ and of type $2^{16}4^6$ which are Gray images of self-dual negacyclic codes over $R$ of length $14$ are listed.Comment: arXiv admin note: text overlap with arXiv:1710.0923

Linear Codes over Z_4+uZ_4: MacWilliams identities, projections, and formally self-dual codes

Linear codes are considered over the ring Z_4+uZ_4, a non-chain extension of Z_4. Lee weights, Gray maps for these codes are defined and MacWilliams identities for the complete, symmetrized and Lee weight enumerators are proved. Two projections from Z_4+uZ_4 to the rings Z_4 and F_2+uF_2 are considered and self-dual codes over Z_4+uZ_4 are studied in connection with these projections. Finally three constructions are given for formally self-dual codes over Z_4+uZ_4 and their Z_4-images together with some good examples of formally self-dual Z_4-codes obtained through these constructions.Comment: 12 pages. Partially presented in the 13th International Workshop on Algebraic and combinatorial coding theory, Pomorie, Bulgaria, 201

Self-dual cyclic codes over M2(Z4)M_2(\mathbb{Z}_4)

In this paper, we study the codes over the matrix ring over $\mathbb{Z}_4$, which is perhaps the first time the ring structure $M_2(\mathbb{Z}_4)$ is considered as a code alphabet. This ring is isomorphic to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ is a root of the irreducible polynomial $x^2+x+1 \in \mathbb{Z}_2[x]$ and $U\equiv$ ${11}\choose{11}$. We first discuss the structure of the ring $M_2(\mathbb{Z}_4)$ and then focus on algebraic structure of cyclic codes and self-dual cyclic codes over $M_2(\mathbb{Z}_4)$. We obtain the generators of the cyclic codes and their dual codes. Few examples are given at the end of the paper.Comment: 10 page

Cyclic codes over Z4+uZ4\mathbb{Z}_4+u\mathbb{Z}_4

In this paper, we have studied cyclic codes over the ring $R=\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2=0$. We have considered cyclic codes of odd lengths. A sufficient condition for a cyclic code over $R$ to be a $\mathbb{Z}_4$-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 generated cyclic codes of odd length over $R$. We have determined a necessary condition and a sufficient condition for cyclic codes of odd lengths over $R$ to be $R$-free.Comment: arXiv admin note: text overlap with arXiv:1412.375

On a class of constacyclic codes over the non-principal ideal ring Zps+uZps\mathbb{Z}_{p^s}+u\mathbb{Z}_{p^s}

$(1+pw)$-constacyclic codes of arbitrary length over the non-principal ideal ring $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are studied, where $p$ is a prime, $w\in \mathbb{Z}_{p^s}^{\times}$ and $s$ an integer satisfying $s\geq 2$. First, the structure of any $(1+pw)$-constacyclic code over $\mathbb{Z}_{p^s} +u\mathbb{Z}_{p^s}$ are presented. Then enumerations for the number of all codes and the number of codewords in each code, and the structure of dual codes for these codes are given, respectively. Then self-dual $(1+2w)$-constacyclic codes over $\mathbb{Z}_{2^s} +u\mathbb{Z}_{2^s}$ are investigated, where $w=2^{s-2}-1$ or $2^{s-1}-1$ if $s\geq 3$, and $w=1$ if $s=2$