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

Due to the rise of social networking, an increasing amount of digital data has been produced in recent years. It is estimated that almost 491 exabytes (EB) of data per day will be accumulated around the globe by 2025. Due to this quick increase of data, it is crucial to store this enormous amount of data in a more effective and efficient manner. Therefore, it is important to find a new medium with high storage capacity and lower energy consumption. Notice that one gram of DNA contains \(2.1\times 10^{9}\) DNA bases. We know that there are four DNA bases and every DNA bases can encode two bits. Hence, one gram of DNA is able to store around \(4.2\times 10^{21}\) bits. On the other hand, conventional storage techniques can store maximum \(10^{9}\) bits. Hence, synthetic DNA is an ideal medium to store a huge amount of data.

In 1994, Adleman [5] solved a combinatorial problem using the structure of DNA, where he used DNA molecules to solve a seven node instance of the famous directed salesman problem. The approach of the above solution was based on the Watson-Crick complement (WCC) property of a DNA strand. Adleman et al. [6] invented a molecular program for breaking the Data Encryption Standard (DES) algorithm. Moreover, in [24] it is shown that DNA molecules can be used as a storage medium. These developments require several theories for the construction of DNA sequences satisfying various constraints. Algebraic coding theory plays an important role in constructing DNA codes with constraints (see [21] for DNA codes over various rings). DNA codes based on error-correcting codes have been successful in DNA-based computation and storage. For example, Milenkovic and Kashyap [23] described the design of codes for DNA computing such that they are avoiding the formation of secondary structure in single-stranded DNA molecules and non-selective cross-hybridization. In the theory of error-correcting codes, cyclic codes form an important class of codes due to their rich algebraic structure and play a significant role in algebraic coding theory. The study of codes over finite rings proliferated after the emergence of Gray maps. In this field, revolution came in 1994 when Hammons et al. [18] obtained some excellent classes of nonlinear binary codes as the images of linear codes over \(\mathbb Z_4.\) This brings the study of cyclic codes over finite rings under immense consideration. The study of cyclic codes over finite rings has been discussed in a series of papers (see [1‐3, 19, 34] and references therein). Some known methods for designing DNA codes that satisfy certain conditions include the study of reversible codes, which constitute an integral part of DNA codes. The study of reversible cyclic codes over finite fields was initiated by Massey [25] in 1964. Later on, Tzeng and Hartmann [33] obtained the bounds for the minimum distance of certain reversible cyclic codes. Guenda and Gullivor [16] discussed reversible cyclic codes over the ring \({\mathbb {F}}_{2}+u{\mathbb {F}}_{2}\) and provided some DNA codes. Moreover, Srinivasulu and Bhaintwal [32] constructed reversible cyclic codes over \(\mathbb F_4+u\mathbb F_4\), \(u^2=0,\) and obtained certain DNA codes. For the intensive study of reversible codes over different rings, we refer the readers to [4, 12, 26, 28‐30, 36].

Although codes over many different rings have been considered, codes over \({\mathbb {Z}}_{4}\) have special place in coding theory due to their connection to lattice designs and DNA codes (Note that \({\mathbb {Z}}_{4}\) being an alphabet of size four, it is an ideal candidate for DNA coding). DNA cyclic codes which are based on \({\mathbb {Z}}_{4}\) have been discussed in a series of papers (see [3, 11, 20] and references therein). Furthermore, D’Yachkov et al. [9, 10] studied the deletion similarity distance between any two codewords. Siap et al. [31] obtained cyclic DNA codes over the ring \({\mathbb {F}}_{2}[u]/\langle u^{2}-1\rangle\) based on the deletion distance. Similarly, Dinh et al. [11] constructed cyclic DNA codes over the ring \({\mathbb {Z}}_4[u]/\langle u^2-1\rangle\) based on the deletion distance. Recently, Martinez-Moro and Szabo [22] classified the structure of local Frobenius non-chain rings of order 16. Later on, Dougherty et al. [13] obtained the structure of cyclic codes over local Frobenius non-chain rings of order 16.

Reversible codes have versatile applications in the design of DNA codes as well as data storage and retrieval systems. If the bandwidth of a communication system increases, we have the ability to transmit a larger amount of data in a single instance. In DNA data storage, the digital data is translated into DNA strings (constituted by four symbols A, G, T and C). For example, two binary bits are mapped into a single nucleotide. That is, \(00\rightarrow A,\) \(01\rightarrow C,\) \(10\rightarrow G\) and \(11\rightarrow T.\) Thus, DNA k-bases can be used instead of DNA single base to carry more data on a wide bandwidth. Constructing a bijection between the elements of the algebraic structure (fields, rings etc.) and DNA k-bases is an important problem. Employing DNA k-bases \((k\ge 2)\) for an algebraic structure uncovers the reversibility problem. For example, if we construct a correspondence between an algebraic structure having only four elements and DNA nucleotides i.e., \(\{A,G,T,C\},\) then a reversible code over this algebraic structure provides a reversible DNA code. However, if an algebraic structure contains \(4^{k}\) elements, where \(k>1,\) then the reversibility problem occurs. This reversibility problem has been addressed by several authors over different algebraic structures. For example, Gursoy et al. [17] solved the reversibility problem using skew cyclic codes over a finite non-chain ring. Furthermore, Oztas et al. [27] identified DNA k-bases with elements in the ring \(\mathbb F_2[u]/\langle u^{2k}-1\rangle ,\) and solved the reversibility problem by using a form of coterm polynomials.

Inspired by these studies, we construct reversible cyclic codes of odd lengths over the ring \({\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle .\) As an application of these reversible codes, we solve the reversibility problem that occurred in DNA k-bases by constructing a bijection \(\Gamma\) (see Table 1) between the elements of the ring \({\mathfrak {R}}\) and \(S_{D_{16}},\) where \(S_{D_{16}}=\{x_{1}x_{2}:x_{1},x_{2}\in {\textbf {X}}\},\) and \({\textbf {X}}\) is the set \(\{A,G,T,C\}.\) Furthermore, we obtain reversible DNA codes over the ring \({\mathfrak {R}}\) based on the deletion distance. Also, we obtain the generating set of dual of cyclic codes over the ring \({\mathfrak {R}}\) and discuss reversibility for these codes. Dougherty et al. [13] constructed binary cyclic codes as the images of Gray map with respect to the Lee weight. In this article, we introduce the homogeneous weight \(w_{\hom },\) and study cyclic codes with respect to \(w_{\hom }\). Meanwhile, we discuss the GC-content of cyclic DNA codes and investigate their deletion distance.

The present article is organized as follows: Sect. 2 is devoted to familiarize the readers with basic terminology. In Sect. 3, we introduce the homogeneous weight over the ring \({\mathfrak {R}}\) and obtain a distance preserving map \(\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}^{8n}_{2}.\) The structure of cyclic codes is conferred in Sect. 4. In Sect. 5, we provide some necessary and sufficient conditions for: (i) a given cyclic code of odd length over the ring \({\mathfrak {R}}\) to be a reversible code, (ii) the dual code of a cyclic code of odd length to be a reversible code over the ring \({\mathfrak {R}},\) and (iii) a given cyclic code of odd length over the ring \({\mathfrak {R}}\) to be reversible complement. The study of GC-content of cyclic DNA codes of odd length and their deletion distance is included in Sect. 6. Section 7 consists examples of reversible DNA codes. The Gray images of some reversible cyclic codes have the same parameters as their corresponding optimal linear codes, which can be found in the online data available in [14].

In [22], the authors provided a complete classification of local Frobenius rings of order 16. Let us recall some basic facts of the ring \({\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle .\) The ring \({\mathfrak {R}}\) is a finite commutative ring, which is an extension of \({\mathbb {Z}}_{4}.\) The ring \({\mathfrak {R}}\) is a vector space over \({\mathbb {F}}_{2}\) with basis \(\{1,u,v,2\}.\) Hence, any element \(x\in {\mathfrak {R}}\) can be uniquely expressed as \(x=\alpha _{1}+\alpha _{2}u+\alpha _{3}v+\alpha _{4}2,\) where \(\alpha _{i}\in {\mathbb {F}}_{2}.\) Moreover, cardinalty of \({\mathfrak {R}}\) is 16 and characteristic of \({\mathfrak {R}}\) is 4. All the elements of the ring \({\mathfrak {R}}\) are as follows:Further, the non-trivial ideals of \({\mathfrak {R}}\) are as follows:The ideal lattice of the ring \({\mathfrak {R}}\) is shown in Fig. 1. Notice that the ideals of \({\mathfrak {R}}\) do not form a chain under the set theoretical inclusion relation. Hence \({\mathfrak {R}}\) is a non-chain ring. Moreover, the ideal \(\langle u,v\rangle\) is the unique maximal ideal of \({\mathfrak {R}}.\) Thus, \({\mathfrak {R}}\) is a local Frobenius non-chain ring of order 16.

Recall that a linear code \({\mathfrak {C}}\) of length n over \({\mathfrak {R}}\) is an \({\mathfrak {R}}\)-submodule of \({\mathfrak {R}}^{n}\) and members of \({\mathfrak {C}}\) are known as codewords. A linear code \({\mathfrak {C}}\) of length n over \({\mathfrak {R}}\) is called cyclic if for each codeword \(x=(x_0,x_1,\ldots ,x_{n-1})\in {\mathfrak {C}},\) the n-tuple \((x_{n-1},x_0,\ldots ,x_{n-2})\) obtained by the cyclic shift of coordinates i to \(i+1\mod ~n,\) is also in \({\mathfrak {C}}\). The inner-product of two n-tuples \(a=(a_0,a_1,\ldots ,a_{n-1})\), \(b=(b_0,b_1,\ldots ,b_{n-1})\) is defined as \(a\cdot b=\sum \nolimits _{i=0}^{n-1}a_ib_i.\) The codewords a and b are said to be orthogonal if \(a\cdot b=0.\)

The dual code \({\mathfrak {C}}^\perp\) of a linear code \({\mathfrak {C}}\) over \({\mathfrak {R}}\) of length n is defined as follows:It is easy to check that \({\mathfrak {C}}^\perp\) is also a linear code of same length n over \({\mathfrak {R}}\). Moreover, a linear code \({\mathfrak {C}}\) is self-orthogonal if \({\mathfrak {C}}\subseteq {\mathfrak {C}}^\perp\) and \({\mathfrak {C}}\) is self-dual if \({\mathfrak {C}}={\mathfrak {C}}^\perp\). For a given polynomial \(g(x)=a_{0}+a_{1}x+\cdots +a_{n-1}x^{n-1}\in {\mathfrak {R}}[x]\), we define the reciprocal of g(x) as \(g^*(x)=x^{\deg (g(x))}g(\frac{1}{x}) =a_{n-1}+a_{n-2}x+\cdots +a_{0}x^{n-1}\). Note that for a given polynomial g(x), \(\deg (g(x))\ge \deg (g^*(x))\) and if \(a_{0}\ne 0\), then \(\deg (g(x))=\deg (g^*(x)).\) Moreover, the polynomial g(x) is called self-reciprocal if \(g^{*}(x)=\alpha g(x),\) where \(\alpha\) is an unit element of \({\mathfrak {R}}.\) Throughout this article, we denote by \({\mathfrak {R}}_{n}\) the quotient ring \({\mathfrak {R}}[x]/\langle x^n-1\rangle .\)

The following lemma is useful to find the reciprocal of sum and product of the polynomials.

Deoxyribonucleic acid abbreviated as DNA is an organic chemical of complex molecular structure that is found in all prokaryotic and eukaryotic cells and in many viruses. DNA codes carry the genetic information of living organisms. Any DNA structure containing nucleotide bases Cytosine (C), Thymine (T) and Adenine (A), Guanine (G) attached to a backbone of alternating phosphate and deoxyribose sugar. Two sugar phosphate chains are paired through hydrogen bonds between G and C and between A and T thus making the twin stranded double helix of the DNA molecules and are joined by the Watson-Crick complement (WCC) rule. According to this rule a DNA sequence combined with its complement (reversible) and forms a helix. Complement of T is A (and vice versa) and complement of G is C (and vice versa). We use \({C}^{c}=G\), \({G}^{c}=C\), \({A}^{c}=T\), \({T}^{c}=A\), where \({C}^{c}\), \({G}^{c}\), \({A}^{c}\), \({T}^{c}\) are the complements of C,  G,  A,  T respectively. For example, the Watson-Crick complement of \(3^{'}-CCATTGA-5^{'}\) is given by \(5^{'}-GGTAACT-3^{'}.\)

We call a given sequence \(x=x_{1}x_{2}\cdots x_{n}\) a quaternary n-sequence or a DNA sequence of length n if \(x_{i}\in {\textbf {X}},\) where \(1\le i\le n.\) Let U and V be any two DNA sequences of length n. The hybridization energy E(U, V) between the two DNA sequences is measured by the longest common sub sequence (not necessarily contiguous) of either strand or the reverse complement of the other strand. We have the deletion similarity as the length of the longest common subsequence for U and V and is denoted by S(U, V). If U and V are any strands of length n,  then we have \(S(U,U)=n\) and \(S(U,V)=S(V,U).\) Moreover, the deletion similarity S(U, V) can be identified by the number of base pair bonds between U and \(V^{rc}\) i.e.,where \(V^{rc}=x^{c}_{n}x^{c}_{n-1}\cdots x^{c}_{1}\) is the reverse complement of \(V=x_{1}x_{2}\cdots x_{n}.\) For example, if \(U=TACAGG\) and \(V=TCAGTT,\) then the deletion similarity \(S(U,V)=3\) as TAG is the longest common subsequence for both U and V. Moreover, TAG is not unique since TCG is another common subsequence of length 3.

By using Definition 2.2, and Condition 2.1, the hybridization energy is given by \(E(U,V^{rc})\le n-D-1\) for all \(U,V\in {\mathfrak {C}},\) \(U\ne V.\) Therefore, in every (n, D)-code any strand U and the reverse complement of the other strand V always form such base pair bonds in hybridization assay which are less than\(n-D.\) From now on, we call D the deletion distance of a code \({\mathfrak {C}}.\)

Moreover, a cyclic DNA code \({\mathfrak {C}}\) always satisfies the Hamming distance constraint, but it may satisfy the other constraints which are as follows: The first three constraints are designed to prevent the unwanted hybridization. The constant GC-content constraint ensures that the thermodynamic properties of each codeword are same.

Constantinescu and Heise [8] introduced the idea of homogeneous weight over the residue class rings of integers. Later on, Greferath and Schmidt [15] generalized this notion to arbitrary finite rings and they have provided the existence of such a weight function on arbitrary finite rings. Dougherty et al. [13] studied cyclic codes over the ring \({\mathfrak {R}}\) via Gray images with respect to the Lee weight. In this article, we study the cyclic codes over the ring \({\mathfrak {R}}\) using the homogeneous weight on \({\mathfrak {R}}.\)

Yildiz and Karadeniz [34] studied cyclic codes over the non-chain ring \(R={\mathbb {F}}_2+u{\mathbb {F}}_2+v{\mathbb {F}}_2+uv{\mathbb {F}}_2,\) where \(u^{2}=0, v^{2}=0\) and \(uv=vu\) with respect to the homogeneous weight. Later on, Yildiz and Kelebek [35] studied homogeneous weight for an infinite family of rings \(R_{k}={\mathbb {F}}_2[u_{1},u_{2},\cdots ,u_{k}]/\langle u^{2}_{i},u_{i}u_{j}-u_{j}u_{i}\rangle .\) Motivated by these studies, in this article we introduce homogeneous weight over the ring \({\mathfrak {R}}\) as follows:

Let \(x\in {\mathfrak {R}}\) be any element. Then we define,Now, in order to define a distance preserving Gray map \(\varphi\) from \({\mathfrak {R}}\) to \({\mathbb {F}}_{2}^{8}\) we let \(\varphi (0)=(0,0,0,0,0,0,0,0),\) \(\varphi (1)=(1,0,1,0,1,0,1,0),\) \(\varphi (u)=(1,1,1,1,0,0,0,0),\) \(\varphi (v)=(1,1,0,0,1,1,0,0)\) and \(\varphi (2)=(1,1,1,1,1,1,1,1).\) We extend this map to the whole ring \({\mathfrak {R}}\) in the following manner. Notice that any \(x\in {\mathfrak {R}}\) can be expressed uniquely as \(x=\alpha _{1}+\alpha _{2}u+\alpha _{3}v+\alpha _{4}2,\) where \(\alpha _{i}\in {\mathbb {F}}_{2}.\) Then, we defineWe can verify that the map \(\varphi :{\mathfrak {R}}\rightarrow {\mathbb {F}}_{2}^{8}\) defined above is a distance preserving map (Gray map). Notice that \(\varphi (1+3)\ne \varphi (1)+\varphi (3),\) therefore we can conclude that the Gray map \(\varphi\) is not linear. Moreover, \(\varphi\) can be extended to the map \(\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}_{2}^{8n}\) such thatwhere \((\beta _{0},\beta _{1},\beta _{2},\ldots ,\beta _{n-1})\in {\mathfrak {R}}^{n}.\) Recall that a cyclic shift on \({\mathfrak {R}}^{n}\) is the permutation \(\tau\) given byA code \({\mathfrak {C}}\) is said to be an \(\ell\)-quasi-cyclic code if it is invariant under \(\tau ^{\ell }\) and we call \(\ell\) the index of the quasi-cyclic code.

Based on the above lemma, we have the following result.

The study of cyclic codes of any odd length over a local Frobenius non-chain ring of order 16 has been developed in [13]. In this article, the aforementioned ring \({\mathfrak {R}}\) is also a local Frobenius non-chain ring of order 16, and hence we obtain the structure of cyclic code over \({\mathfrak {R}}\) using results given in [13]. Throughout this article, we use a symbol M, where M is the maximal ideal of the ring \({\mathfrak {R}}\) given by \(M=\langle u,v\rangle ,\) and \(Mg(x)=\langle ug(x),vg(x)\rangle ,\) where \(g(x)\in {\mathfrak {R}}_{n}.\)

The following theorem provides the structure of cyclic codes of odd lengths over the ring \({\mathfrak {R}}.\)

Notice that in the above theorem, there is no relation between the generating polynomials of cyclic codes. Using the above structure of cyclic codes over the ring \({\mathfrak {R}},\) we obtain the following equivalent structure of cyclic codes in which generating polynomials are dividing some other generating polynomials.

Now, by using the structure of cyclic codes given in Theorem 4.2, we focus on the following specific cases of cyclic codes over the ring \({\mathfrak {R}}.\)

Next, we discuss the annihilator \(A({\mathcal {I}})\) of an ideal \({\mathcal {I}}\) of \({\mathfrak {R}}_{n}\), which is defined as follows:It is easy to check that \(A({\mathcal {I}})\) is an ideal of \({\mathfrak {R}}_{n}\) and hence a cyclic code over \({\mathfrak {R}}\). Moreover, if \({\mathcal {I}}\) generates a cyclic code \({\mathfrak {C}}\) of length n over \({\mathfrak {R}}\), then the ideals which generate \({\mathfrak {C}}^{\perp }\) are given by \(A^{*}({\mathcal {I}})=\{h^{*}(x)~|~h(x)\in A({\mathcal {I}})\},\) where \(h^{*}(x)\) is the reciprocal polynomial of h(x).

By using the next result one can obtain the structure of annihilator of cyclic codes \({\mathfrak {C}}_{i}\) over the ring \({\mathfrak {R}},\) where \(i\in \{1,2,3\}.\)

The next theorem provides the structure of the dual of a cyclic code \({\mathfrak {C}}_{i}\) over the ring \({\mathfrak {R}},\) where \(i\in \{1,2,3\}.\)

In this section, we present our main results of this article. We provide necessary and sufficient conditions for cyclic codes and their duals to be reversible. Also, we discuss under what conditions a given cyclic code is reversible complement, and construct a bijection \(\Gamma\) between the elements of the ring \({\mathfrak {R}}\) and \(S_{D_{16}}\) in such a manner that the reversibility problem is solved.

The next theorem provides the necessary and sufficient conditions for a given cyclic code \({\mathfrak {C}}\) over the ring \({\mathfrak {R}}\) to be reversible.

The following result provides the reversibility conditions for cyclic codes \({\mathfrak {C}}_{i},\) where \(i\in \{1,2,3\}.\)

The next result is useful to obtain the reversibility of the dual of cyclic codes \({\mathfrak {C}}_{i},\) where \(i\in \{1,2,3\}.\)

Next, we discuss the reversible complement cyclic codes of odd length and provide a necessary and sufficient condition for a given cyclic code to be a reversible complement cyclic code over the ring \({\mathfrak {R}}.\) To do so, we need the following basic notions of complement. We denote the complement of an element \(a\in {\mathfrak {R}}\) by \(\bar{a}\) and define as: \(\bar{a}=a+2\). For example, the complement of \(a=3+u+v\) is \(\bar{a}=3+u+v+2=1+u+v.\) Recall that the Watson-Crick complement for a given DNA sequence \(X=d_{1}d_{2}\cdots d_{n}\) is given by \({X}^{c}=d^{c}_{1} d^{c}_{2}\cdots d^{c}_{n}\). From Table 1, we notice that if \(d_{1}d_{2}\) is the DNA sequence corresponding to an element \(x\in {\mathfrak {R}},\) then \(d^{c}_{1} d^{c}_{2}\) corresponds to the element \(x+2,\) where \(d^{c}_{1} d^{c}_{2}\) is the Watson-Crick complement of \(d_{1}d_{2}.\) Similarly, for an element \(g(x)\in {\mathfrak {R}}[x],\) where \(g(x)=\alpha _{0}x+\alpha _{1}x+\alpha _{2}x^{2}+\cdots +\alpha _{n-1}x^{n-1}\), the complement of g(x) is given bywhere \(\bar{\alpha }_{i}\) is the complement of \(\alpha _{i}\) for \(\alpha _{i}\in {\mathfrak {R}}\), \(0\le i\le n-1\). The reverse complement of \(g(x)\in {\mathfrak {R}}[x]\) is denoted by \(g^{rc}(x),\) and is defined asIn Table 1, we have constructed a bijection \(\Gamma\) between the elements of the ring \({\mathfrak {R}}\) and \(S_{D_{16}}\) in such a manner that the reversibility problem is solved. We state the reversibility problem as follows:

Reversibility Problem: Let \({\mathcal {R}}\) be a ring with cardinality \(4^{k},\) where \(k\ge 2.\) Suppose that there exists a bijection \(\vartheta\) between the elements of the ring \({\mathcal {R}}\) and DNA k-bases. If \({\mathcal {C}}\) is a reversible cyclic code over the ring \({\mathcal {R}}\), then \(\vartheta ({\mathcal {C}})\) is not necessarily a reversible DNA cyclic code.

In order to clarify the above problem, we present an example. Let \(\lambda =(\lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4})\) be any codeword over the ring \({\mathfrak {R}},\) where \(\lambda _{i}\in {\mathfrak {R}}.\) Also, suppose that ATCGGATG is the DNA sequence corresponding to the codeword \(\lambda =(\lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4})\) such that \(\lambda _{1}\) corresponds to AT,  \(\lambda _{2}\) corresponds to CG,  \(\lambda _{3}\) corresponds to GA and \(\lambda _{4}\) corresponds to TG. Now, the reverse of \(\lambda\) is given by \(\lambda ^{r}=(\lambda _{4},\lambda _{3},\lambda _{2},\lambda _{1})\) and corresponding DNA sequence is TGGACGAT. Notice that TGGACGAT is not the reverse of the DNA sequence ATCGGATG,  the reverse of ATCGGATG is GTAGGCTA. We solved this reversibility problem by making use of the correspondence given in the Table 1. Hence, we present the following lemma which plays a key role in solving the reversibility problem.

Now, we provide a necessary and sufficient condition under which a given cyclic code \({\mathfrak {C}}_{i},\) where \(i\in \{1,2,3\}\) over the ring \({\mathfrak {R}}\) of any odd length is a reversible complement cyclic code.

Let \(x=x_{1}x_{2}\cdots x_{n}\) be any DNA sequence of length n. Then the GC-content of x is defined to be the number of indices i such that \(x_{i}\in \{G,C\}.\) The DNA is a long chain of nucleotides. Each base binds with another complementary base via hydrogen bonds. For example, A forms two hydrogen bonds with T, (similarly T forms two hydrogen bonds with A) and G forms three hydrogen bonds with C, (similarly C forms three hydrogen bonds with G). Moreover, in DNA codes the same GC-content in every codeword ensures that the codewords have similar hybridization energy and melting temperature. This motivates considering such DNA codes in which all codewords have the same GC-content. In this section, we study the GC-content of a DNA cyclic code over the ring \({\mathfrak {R}}\) and deletion distance D of such DNA codes.

Let \({\mathfrak {C}}_{i}=\langle \ell _{0}(x),M\ell _{1}(x),\xi _{i}\ell _{2}(x),2\ell _{3}(x)\rangle\) be a cyclic code of odd length n over the ring \({\mathfrak {R}},\) where \(i\in \{1,2,3\}.\) Then we define a subcode \({\mathfrak {C}}^{(2)}_{i}\) of \({\mathfrak {C}}_{i}\) such that \({\mathfrak {C}}^{(2)}_{i}\) consists of all codewords of \({\mathfrak {C}}_{i}\) which are multiples of 2.

The next result provides the structure of \({\mathfrak {C}}^{(2)}_{i},\) where \(i\in \{1,2,3\},\) which will be further useful in obtaining the spectra of GC-content of cyclic codes \({\mathfrak {C}}_{i}\) in Theorem 6.3.

We have the following result which provides the spectra of the GC-content of \({\mathfrak {C}}_{i}\) over the ring \({\mathfrak {R}},\) where \(i\in \{1,2,3\}.\)

The next theorem provides the deletion distance of the subcode \({\mathfrak {C}}^{(2)}_{i},\) where \(i\in \{1,2,3\}.\)

In this section, we present some examples of reversible cyclic codes over the ring \({\mathfrak {R}},\) and their \(\Gamma\) images are reversible DNA codes, which are based on the deletion distance. In Table 2, we provide some examples of reversible cyclic codes and their Gray images.

We have studied reversible cyclic codes of odd lengths over the ring \({\mathfrak {R}}.\) As an application of these reversible cyclic codes, DNA cyclic codes of odd length over the ring \({\mathfrak {R}}\) based on the deletion distance have been obtained. Moreover, we have constructed DNA codes which satisfy the Hamming constraint, the reverse and the reversible complement constraint. The GC-content and their deletion distance of these codes are also discussed. We have also established a relation between the GC-content of a given cyclic code \({\mathfrak {C}}_{i}\) and its subcode \({\mathfrak {C}}^{(2)}_{i}.\) Also, we have introduced a Gray map \(\Psi _{\hom }\) from (\({\mathfrak {R}}^{n}\), Homogeneous weight) to (\({\mathbb {F}}^{8n}_{2}\), Hamming weight). Meanwhile, we have constructed a bijection \(\Gamma\) in such a way that we have solved the reversibility problem. In the end, we have provided examples of DNA cyclic codes with their deletion distances. Furthermore, we have provided some examples of reversible cyclic codes and their Gray images have same parameters as their corresponding optimal linear codes. For future work, it would also be fascinating to construct reversible cyclic codes of even lengths over the ring \({\mathfrak {R}}.\)

Remark All codes in the above examples are calculated using Magma software [7], and * delineates that the corresponding binary linear code is optimal with respect to the online database of linear code [14].