Structure of linear codes over the ring BkB_k

We study the structure of linear codes over the ring $B_k$ which is defined by $\mathbb{F}_{p^r}[v_1,v_2,\ldots,v_k]/\langle v_i^2=v_i,~v_iv_j=v_jv_i \rangle_{i,j=1}^k.$ In order to study the codes, we begin with studying the structure of the ring $B_k$ via a Gray map which also induces a relation between codes over $B_k$ and codes over $\mathbb{F}_{p^r}.$ We consider Euclidean and Hermitian self-dual codes, MacWilliams relations, as well as Singleton-type bounds for these codes. Further, we characterize cyclic and quasi-cyclic codes using their images under the Gray map, and give the generators for these type of codes.Comment: 18 pages, accepted for publication (Journal of Applied Mathematics and Computing

Codes over an algebra over ring

In this paper, we consider some structures of linear codes over the ring $\mathcal{R}_k=R[v_1,\dots,v_k],$ where $v_i^2=v_i$ forall $i=1,\dots,k),$ and $R$ is a finite commutative Frobenius ring

Constacyclic codes over F_q + u F_q + v F_q + u v F_q

Let q be a prime power and F_q be a finite field. In this paper, we study constacyclic codes over the ring F_q+ u F_q +v F_q+ u v F_q, where u^2=u, v^2=v and uv=vu. We characterized the generator polynomials of constacyclic codes and their duals using some decomposition of this ring. We also define a gray map and characterize the Gray images of self-dual cyclic codes over F_q+uF_q+vF_q+uvF_q

ΘS−\Theta_S-cyclic codes over AkA_k

We study $\Theta_S-$cyclic codes over the family of rings $A_k.$ We characterize $\Theta_S-$cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is $\Theta_S-$cyclic then its Hermitian dual is also $\Theta_S-$cyclic. Finally, we give constructions of $\Theta_S-$cyclic codes.Comment: 23 page

Skew-Cyclic Codes over BkB_k

In this paper we study the structure of $\theta$-cyclic codes over the ring $B_k$ including its connection to quasi-$\tilde{\theta}$-cyclic codes over finite field $\mathbb{F}_{p^r}$ and skew polynomial rings over $B_k.$ We also characterize Euclidean self-dual $\theta$-cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean $\theta$-cyclic codes

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

The Art of DNA Strings: Sixteen Years of DNA Coding Theory

The idea of computing with DNA was given by Tom Head in 1987, however in 1994 in a seminal paper, the actual successful experiment for DNA computing was performed by Adleman. The heart of the DNA computing is the DNA hybridization, however, it is also the source of errors. Thus the success of the DNA computing depends on the error control techniques. The classical coding theory techniques have provided foundation for the current information and communication technology (ICT). Thus it is natural to expect that coding theory will be the foundational subject for the DNA computing paradigm. For the successful experiments with DNA computing usually we design DNA strings which are sufficiently dissimilar. This leads to the construction of a large set of DNA strings which satisfy certain combinatorial and thermodynamic constraints. Over the last 16 years, many approaches such as combinatorial, algebraic, computational have been used to construct such DNA strings. In this work, we survey this interesting area of DNA coding theory by providing key ideas of the area and current known results.Comment: 19 pages, 4 figures, draft review on DNA code

Constacyclic Codes Over Finite Principal Ideal Rings

In this paper, we give an important isomorphism between contacyclic codes and cyclic codes over finite principal ideal rings. Necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite principal ideal rings are given

The zero short Covering Problem for finite rings

In this work, we find the cardinality of minimal zero short covers of An for any finite local ring A, removing the restriction of D(A)^2 = 0 from the previous works in the literature. Using the structure theorem for Artinian rings, we conclude that we have solved the zero short covering problem for all finite rings. We demonstrate our results on R_k, an infinite family of finite commutative rings extensively studied in coding theory, which satisfy D(A)^2 \neq 0 for all k \geq 2.Comment: This paper has been withdrawn by the author due to incorrect calculations and conclusion

On the codes over the Z_3+vZ_3+v^2Z_3

In this paper, we study the structure of cyclic, quasi-cyclic, constacyclic codes and their skew codes over the finite ring R=Z_3+vZ_3+v^2Z_3, v^3=v. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic and skew constacyclic codes over R are obtained. A necessary and sufficient condition for cyclic (negacyclic) codes over R that contains its dual has been given. The parameters of quantum error correcting codes are obtained from both cyclic and negacyclic codes over R. It is given some examples. Firstly, quasi-constacyclic and skew quasi-constacyclic codes are introduced. By giving two product, it is investigated their duality. A sufficient condition for 1-generator skew quasi-constacyclic codes to be free is determined