On Cyclic DNA Codes

This paper considers cyclic DNA codes of arbitrary length over the ring R=\F_2[u]/u^4-1. A mapping is given between the elements of $R$ and the alphabet $\{A,C,G,T\}$ which allows the additive stem distance to be extended to this ring. Cyclic codes over $R$ are designed such that their images under the mapping are also cyclic or quasi-cyclic of index 2. The additive distance and hybridization energy are functions of the neighborhood energy.Comment: there is an error in Lemma 3.

On cyclic DNA codes over the Ring Z4+uZ4\Z_4 + u \Z_4

In this paper, we study the theory for constructing DNA cyclic codes of odd length over $\Z_4[u]/\langle u^2 \rangle$ which play an important role in DNA computing. Cyclic codes of odd length over $\Z_4 + u \Z_4$ satisfy the reverse constraint and the reverse-complement constraint are studied in this paper. The structure and existence of such codes are also studied. The paper concludes with some DNA example obtained via the family of cyclic codes.Comment: 16 page

On cyclic DNA codes over F2+uF2+u2F2\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2

In the present paper we study the structure of cyclic DNA codes of even lenght over the ring $\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ where $u^3=0$. We investigate two presentations of cyclic codes of even lenght over $\mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2$ satisfying the reverse constraint and reverse-complement constraint.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1508.02015 by other author

On DNA Codes using the Ring Z4 + wZ4

In this work, we study the DNA codes from the ring R = Z4 + wZ4, where w^2 = 2+2w with 16 elements. We establish a one to one correspondence between the elements of the ring R and all the DNA codewords of length 2 by defining a distance-preserving Gau map phi. Using this map, we give several new classes of the DNA codes which satisfies reverse and reverse complement constraints. Some of the constructed DNA codes are optimal.Comment: Revised version with new results and corrections. Submitted to ISIT 201

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

Construction of cyclic DNA codes over the Ring Z4[u]/⟨u2−1⟩\Z_4[u]/\langle u^2-1 \rangle Based on the deletion distance

In this paper, we develop the theory for constructing DNA cyclic codes of odd length over $R=\Z_4[u]/\langle u^2-1 \rangle$ based on the deletion distance. Firstly, we relate DNA pairs with a special 16 elements of ring $R$. Cyclic codes of odd length over $R$ satisfy the reverse constraint and the reverse-complement constraint are discussed in this paper. We also study the $GC$-content of these codes and their deletion distance. The paper concludes with some examples of cyclic DNA codes with $GC$-content and their respective deletion distance.Comment: 15 pages, 3 tables. arXiv admin note: substantial text overlap with arXiv:1508.02015, arXiv:1511.0393

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

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

Greedy Construction of DNA Codes and New Bounds

In this paper, we construct linear codes over $\mathbb{Z}_4$ with bounded $GC$-content. The codes are obtained using a greedy algorithm over $\mathbb{Z}_4$. Further, upper and lower bounds are derived for the maximum size of DNA codes of length $n$ with constant $GC$-content $w$ and edit distance $d$

Reversible DNA codes over a family of non-chain rings

In this work we extend results introduced in [15]. Especially, we solve the reversibility problem for DNA codes over the non chain ring $R_{k,s}=F_{4^{2k}}[u_1,\ldots,u_{s}]/\langle u_1^2-u_1,\ldots, u_s^2-u_s\rangle$. We define an automorphism $\theta$ over $R_{k,s}$ which helps us both finding the idempotent decomposition of $R_{k,s}$ and solving the reversibility problem via skew cyclic codes. Moreover, we introduce a generalized Gray map that preserves DNA reversibility.Comment: 10 page