Reversible Cyclic Codes Research Articles

On reversible DNA codes over the ring $${\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ based on the deletion distance

AbstractLet $${\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where $${\mathbb {Z}}_{4}$$ Z 4 is a ring of integers modulo 4. This ring $${\mathfrak {R}}$$ R is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length n over the ring $${\mathfrak {R}}.$$ R . Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n, based on the deletion distance over the ring $${\mathfrak {R}}.$$ R . We also construct a bijection $$\Gamma$$ Γ between the elements of the ring $${\mathfrak {R}}$$ R and $$S_{D_{16}}.$$ S D 16 . As an application of $$\Gamma ,$$ Γ , the reversibility problem which occurs in DNA k-bases has been solved. Moreover, we introduce a Gray map $$\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}_{2}^{8n}$$ Ψ hom : R n → F 2 8 n with respect to homogeneous weight $$w_{\hom }$$ w hom over the ring $${\mathfrak {R}}$$ R . Further, we discuss the GC-content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.

The Extended Codes of a Family of Reversible MDS Cyclic codes

The Extended Codes of a Family of Reversible MDS Cyclic codes

On reversibility problem in DNA bases over a class of rings

Let R k = Z 4 [ u 1 , u 2 , … , u k ] / ⟨ u i 2 − u i , u i u j − u j u i ⟩ be a non-chain ring of characteristic 4, where 1 ≤ i , j ≤ k and k ≥ 1 . In this article, we discuss reversible cyclic codes of odd lengths over the ring R k . We construct bijections between the elements of the ring R k and DNA- 2 k bases for k = 1, 2 in such a manner that the reversibility problem is solved. Employing these bijections, reversible complement cyclic codes of odd lengths are generated. Furthermore, we construct a Gray map Φ : R k n → Z 4 2 k n and as an application of the Gray map Φ, we obtain the GC-content of cyclic codes of arbitrary odd length over the ring R k . Meanwhile, we provide some examples of reversible cyclic codes of odd lengths over the ring R k for different values of k, and also obtain the Lee distances of these codes.

DNA CODES FROM REVERSIBLE GROUP CODES BY A VIRUS OPTIMISATION ALGORITHM

In this paper, we employ group rings and some known results on group codes to study reversible group DNA codes. We define and study reversible cyclic DNA codes from a group ring point of view and we also introduce the notion for self-reciprocal group ring elements. Moreover, we search for reversible group DNA codes with the use of a virus optimisation algorithm. We obtain many good DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement and the fixed GC-content constraints.

Reversible Cyclic Codes over F2+uF2+u2F2

In this article, we investigate the formation of reversible cyclic codes (i.e., its codewords forms a symmetry) over the ring S=F2+uF2+u2F2, where u3=0. We find a unique set of generators for cyclic codes over S and classify reversible cyclic codes to their generators. The dual reversible cyclic codes are studied as well. Moreover, we provide some examples of reversible cyclic codes.

Reversible complement cyclic codes over ℤ4 + uℤ4 + vℤ4 for DNA computing

In this paper, we develop a theory for constructing cyclic codes of odd length over the ring [Formula: see text], where [Formula: see text], which plays an important role in DNA computing. A direct link between the elements of [Formula: see text] and the [Formula: see text] codons used in the amino acids of the living organisms is established. Then, we investigate reversible cyclic codes and reversible complement cyclic codes of odd length over [Formula: see text]. Moreover, we give some properties of binary images of the codons under the Gray map. Finally, two examples of cyclic codes over [Formula: see text] with their minimum Hamming distance will be studied as well.

Reversible cyclic codes over a class of chain rings and their application to DNA codes

Reversible cyclic codes over a class of chain rings and their application to DNA codes

Reversible cyclic codes over some finite rings and their application to DNA codes

Reversible cyclic codes over some finite rings and their application to DNA codes

On LCD codes over Z 4

Linear complementary dual(LCD) codes are being utilized to protect cryptographic implementations from side-channel and fault attacks. In this work, LCD codes over the ring Z 4 are examined. It is observed that an LCD cyclic code over Z 4 of odd length is always reversible. However, the converse of this result does not hold. A suitable example to validate this statement is presented. Further, a necessary and sufficient condition for a reversible cyclic code of odd length to be an LCD code is obtained. Finally, it is found that among cyclic codes of length 2 k , no non-trivial LCD code exists.

Construction of reversible cyclic codes over ℤ p k

For a prime p and integer k > 1, we find the unique set of generators for cyclic codes over . Besides, the necessary and sufficient conditions for cyclic codes to be reversible are obtained. Finally, as an application, some computational examples are given in support of our results.

Reversible cyclic codes over a class of chain rings and their application to DNA codes

Reversible cyclic codes over a class of chain rings and their application to DNA codes

Constacyclic and Linear Complementary Dual Codes Over Fq + uFq

This article discusses linear complementary dual (LCD) codes over ℜ = Fq+uFq(u2=1) where q is a power of an odd prime p. Authors come up with a new Gray map from ℜn to F2nq and define a new class of codes obtained as the gray image of constacyclic codes over .ℜ Further, we extend the study over Euclidean and Hermitian LCD codes and establish a relation between reversible cyclic codes and Euclidean LCD cyclic codes over ℜ. Finally, an application of LCD codes in multisecret sharing scheme is given.

Efficient and Explicit Balanced Primer Codes

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$ , minimum distance $d$ with $(d+1) \log _{4}\,\,n +O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth’s balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing.

Reversible complement cyclic codes over Galois rings with application to DNA codes

Let R be a Galois ring of characteristic pa and cardinality pam, where p is a prime and a,m are natural numbers. For a unit u and an element k∈R such that u2=1 and uk=k, a generalized notion of a complement of an element e in R with respect to u and k has been given in this paper. Using this notion of complement, reversible complement cyclic codes with respect to u and k have been defined. Further, a necessary and sufficient condition for a reversible cyclic code of arbitrary length over R to be a reversible complement cyclic code with respect to u and k has been obtained. This generalization is significant in view of the fact that a reversible cyclic code of length n over R may be reversible complement with respect to one choice of u and k but may not be reversible complement with respect to some other choice of u and k. Further, reversible complement cyclic codes with respect to u and k (for different values of u and k) over a special Galois ring, Z4, have been used to construct some DNA codes. It has been observed that codes over Z4 in some cases give better DNA codes than DNA codes over fields and some rings of equal cardinality. A tradeoff between the two DNA properties namely avoiding secondary structure formation and constant GC−content has also been observed.

DNA Codes Over the Ring F₄[U]/〈U³〉

In this paper, we develop the method for constructing DNA codes of odd length over the finite chain ring $R=\mathbb {F}_{4}[u]/\langle u^{3}\rangle $ , which plays an important role in genetics, bioengineering and DNA computing. By using a Gray map $\eta $ from $R$ to $\mathbb {F}_{4}$ , we give a one-to-one correspondence between 64 DNA codons of the alphabet $\{A, T, G,C\}^{3}$ and the 64 elements of the chain ring $R$ . We also establish a map $\Theta $ from $R^{n}$ to $\{A, T, G,C\}^{3n}$ . Then we provide a necessary and sufficient condition for cyclic codes of odd length over the chain ring $R$ to be reversible. Finally, when $C$ is a reversible cyclic code over $R$ , we derive a necessary and sufficient condition for $\Theta (C)$ to be a DNA code.

New Characterization and Parametrization of LCD Codes

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their symplectic basis. Using such a characterization,we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results of the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.

DNA computing over the ring ℤ4[v]/〈v2 − v〉

In this paper, we discuss the DNA construction of general length over the finite ring [Formula: see text], with [Formula: see text], which plays a very significant role in DNA computing. We discuss the GC weight of DNA codes over [Formula: see text]. Several examples of reversible cyclic codes over [Formula: see text] are provided, whose [Formula: see text]-images are [Formula: see text]-linear codes with good parameters.

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.

On quantum tensor product codes

We present a general framework for the construction of quantum tensor product codes (QTPC). In a classical tensor product code (TPC), its parity check matrix is constructed via the tensor product of parity check matrices of the two component codes. We show that by adding some constraints on the component codes, several classes of dual-containing TPCs can be obtained. By selecting different types of component codes, the proposed method enables the construction of a large family of QTPCs and they can provide a wide variety of quantum error control abilities. In particular, if one of the component codes is selected as a burst-error-correction code, then QTPCs have quantum multiple-burst-error-correction abilities, provided these bursts fall in distinct subblocks. Compared with concatenated quantum codes (CQC), the component code selections of QTPCs are much more flexible than those of CQCs since only one of the component codes of QTPCs needs to satisfy the dual-containing restriction. We show that it is possible to construct QTPCs with parameters better than other classes of quantum error-correction codes (QECC), e.g., CQCs and quantum BCH codes. Many QTPCs are obtained with parameters better than previously known quantum codes available in the literature. Several classes of QTPCs that can correct multiple quantum bursts of errors are constructed based on reversible cyclic codes and maximum-distance-separable (MDS) codes.

LCD Cyclic Codes Over Finite Fields

In addition to their applications in data storage, communications systems, and consumer electronics, linear complementary dual (LCD) codes—a class of linear codes—have been employed in cryptography recently. LCD cyclic codes were referred to as reversible cyclic codes in the literature. The objective of this paper is to construct several families of reversible cyclic codes over finite fields and analyze their parameters. The LCD cyclic codes presented in this paper have very good parameters in general, and contain many optimal codes. A well rounded treatment of reversible cyclic codes is also given in this paper.