Images Of Cyclic Codes Research Articles

New Quantum Codes from the Images of Cyclic Codes

New Quantum Codes from the Images of Cyclic Codes

New quantum codes derived from images of cyclic codes

New quantum codes derived from images of cyclic codes

Cyclic and constacyclic codes over the ring Z4[u]/⟨u3 − u2⟩ and their Gray images

In this article, the structure of generator polynomial of the cyclic codes with odd length is formed over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4}$ where $u^3=u^2$. With the isomorphism we have defined, the generator polynomial of constacyclic codes with odd length over this ring is created from the generator of the cyclic codes. Additionally, necessary and sufficient conditions for a linear code in this ring to be a self dual code and a LCD code are mentioned. Furthermore, for all units over this ring, $\mathbb{Z}_{4}$-images of $\lambda$-constacyclic codes and also $\mathbb{Z}_{4}$-images of cyclic codes are examined by using related ones from defined three new Gray maps. Moreover, several new and optimal codes are constructed in terms of the Lee, Euclidean and Hamming weight in reference to the database.

Cyclic codes over $${\mathbb {F}}_2 +u{\mathbb {F}}_2+v{\mathbb {F}}_2 +v^2 {\mathbb {F}}_2 $$ with respect to the homogeneous weight and their applications to DNA codes

In this paper, we study cyclic codes and their duals over the local Frobenius non-chain ring $$R={\mathbb {F}}_2[u,v] / \langle u^2=v^2,uv \rangle $$, and we obtain optimal binary linear codes with respect to the homogeneous weight over R via a Gray map. Moreover, we characterize DNA codes as images of cyclic codes over R.

DNA cyclic codes over the ring 𝔽2[u,v]/〈u2 − 1,v3 − v,uv − vu〉

In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements [Formula: see text] [Formula: see text] to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring [Formula: see text], which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring [Formula: see text] and 64 codons used in the amino acids of living organisms by introducing a Gray map from [Formula: see text] to [Formula: see text]. Among others, binary images of cyclic codes over [Formula: see text] are also investigated. As applications, some cyclic DNA codes over [Formula: see text] using the Gray map are provided.

Cyclic codes over the ring $$F_2+uF_2+vF_2$$ F 2 + u F 2 + v F 2

In this paper, we study linear and cyclic codes over the ring $$F_2+uF_2+vF_2$$ . The ring $$F_2+uF_2+vF_2$$ is the smallest non-Frobenius ring. We characterize the structure of cyclic codes over the ring $$R=F_2+uF_2+vF_2$$ using of the work Abualrub and Saip (Des Codes Cryptogr 42:273–287, 2007). We study the rank and dual of cyclic codes of odd length over this ring. Specially, we show that the equation $$|C||C^\bot |= |R|^n$$ does not hold in general for a cyclic code C of length n over this ring. We also obtain some optimal binary codes as the images of cyclic codes over the ring $$F_2+uF_2+vF_2$$ under a Gray map, which maps Lee weights to Hamming weights. Finally, we give a condition for cyclic codes over R that contains its dual and find quantum codes over $$F_2$$ from cyclic codes over the ring $$F_2+uF_2+vF_2$$ .

On the structure of cyclic codes over F q + uF q + vF q + uvF q

In this paper, cyclic codes over the ring R = Fq + uFq + vFq + uvFq + are discussed where the ring R is not a finite chain ring. By studying the polynomial ring ℜn = (Fq + uFq + vFq + uvFq)[x/(xn − 1) and using the corresponding relationship between the cyclic codes in R and the ideals in ℜn, cyclic codes over the ring R are characterized. Finally, a Gray-map is obtained and the image of cyclic codes in R is characterized.

Binary Images of Cyclic codes over 4

We determine all linear cyclic codes over Z4 of odd length whose Gray images are linear codes or, equivalently, whose Netchaev-Gray image are linear cyclic codes.