The homogeneous weight for RkR_k, related Gray map and new binary quasicyclic codes

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family $R_k$, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over $R_k$ under this Gray map. We then discuss quasitwisted codes over $R_k$ and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasicyclic. In particular, we find a substantial number of optimal binary codes that are quasicyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasicyclic codes kept by Chen.Comment: Submitted to be publishe

Cyclic DNA codes over F2+uF2+vF2+uvF2

In this work, we study the structure of cyclic DNA codes of arbitrary lengths over the ring R=F2+uF2+vF2+uvF2 and establish relations to codes over R1=F2+uF2 by defining a Gray map between R and R1^2 where R1 is the ring with 4 elements. Cyclic codes of arbitrary lengths over R satisfied the reverse constraint and the reverse-complement constraint are studied in this paper. The GC content constraint is considered in the last

DNA Cyclic Codes Over The Ring \F_2[u,v]/\langle u^2-1,v^3-v,uv-vu \rangle

In this paper, we mainly study the some structure of cyclic DNA codes of odd length over the ring R = \F_2[u,v]/\langle u^2-1,v^3-v,uv-vu \rangle which play an important role in DNA computing. We established a direct link between the element of ring $R$ and 64 codons by introducing a Gray map from $R$ to $R_1 = F_2 + uF_2, u^2 = 1$ where $R_1$ is the ring of four elements. The reverse constrain and the reverse-complement constraint codes over $R$ and $R_1$ are studied in this paper. Binary image of the cyclic codes over R also study. The paper concludes with some example on DNA codes obtained via gray map.Comment: 17 pages, 4 Tables(Table 1 contained 2 pages). arXiv admin note: substantial text overlap with arXiv:1508.02015; text overlap with arXiv:1508.07113, arXiv:1505.06263 by other author

A family of constacyclic codes over F2m+uF2m\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}} and its application to quantum codes

We introduce a Gray map from $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ to $\mathbb{F}_{2}^{2m}$ and study $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}},$ where $u^{2}=0.$ It is proved that the image of a $(1+u)$-constacyclic code length $n$ over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ under the Gray map is a distance-invariant quasi-cyclic code of index $m$ and length $2mn$ over $\mathbb{F}_{2}.$ We also prove that every code of length $2mn$ which is the Gray image of cyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ of length $n$ is permutation equivalent to a binary quasi-cyclic code of index $m.$ Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to $(1+u)$-constacyclic codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}.

Reversible Codes and Its Application to Reversible DNA Codes over F4kF_{4^k}

Coterm polynomials are introduced by Oztas et al. [a novel approach for constructing reversible codes and applications to DNA codes over the ring $F_2[u]/(u^{2k}-1)$, Finite Fields and Their Applications 46 (2017).pp. 217-234.], which generate reversible codes. In this paper, we generalize the coterm polynomials and construct some reversible codes which are optimal codes by using $m$-quasi-reciprocal polynomials. Moreover, we give a map from DNA $k$-bases to the elements of $F_{4^k}$, and construct reversible DNA codes over $F_{4^k}$ by DNA-$m$-quasi-reciprocal polynomials

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 $R$

On Quantum Codes Obtained From Cyclic Codes Over F2+uF2+u2F2\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2

Let $R=\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ be a non-chain finite commutative ring, where $u^3=u$. In this paper, we mainly study the construction of quantum codes from cyclic codes over $R$. We obtained self-orthogonal codes over $\mathbb{F}_2$ as gray images of linear and cyclic codes over $R$. The parameters of quantum codes which are obtained from cyclic code over $R$ are discussed.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1407.1232 by other author

(1−2uk)(1-2u^k)-constacyclic codes over Fp+uFp+u2F+u3Fp+⋯+ukFp\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_+u^{3}\mathbb{F}_{p}+\dots+u^{k}\mathbb{F}_{p}

Let $\mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}=\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_p+u^{3}\mathbb{F}_{p}+\cdots+u^{k}\mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem

Polyadic cyclic codes over a non-chain ring Fq[u,v]/⟨f(u),g(v),uv−vu⟩\mathbb{F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle

Let $f(u)$ and $g(v)$ be any two polynomials of degree $k$ and $\ell$ respectively ($k$ and $\ell$ are not both $1$), which split into distinct linear factors over $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle f(u),g(v),uv-vu\rangle$ be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring $\mathcal{R}$. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from $\mathcal{R}^n \rightarrow \mathbb{F}^{k\ell n}_q$ which preserves duality. The Gray images of polyadic codes and their extensions over the ring $\mathcal{R}$ lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over $\mathbb{F}_q$. Some examples are also given to illustrate this

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