On DNA codes from a family of chain rings ∗

: In this work, we focus on reversible cyclic codes which correspond to reversible DNA codes or reversible-complement DNA codes over a family of ﬁnite chain rings, in an eﬀort to extend what was done by Yildiz and Siap in [20]. The ring family that we have considered are of size 2 2 k , k = 1 , 2 , · · · and we match each ring element with a DNA 2 k − 1 -mer. We use the so-called u 2 -adic digit system to solve the reversibility problem and we characterize cyclic codes that correspond to reversible-complement DNA-codes. We then conclude our study with some examples.


Introduction
In [3], Adleman proposed a solution to an instance of the NP-complete problem of the directed Hamiltonian Path problem, using DNA molecules.This ground breaking approach of using the DNA as a computational tool has led to many similar studies since its appearance.In [6] and [4], the advantages of these studies were demonstrated by a molecular program that led to breaking the Data Encryption System.In [11], Mansuripur et al. show that DNA molecules can be used as a storage medium.
The DNA sequences consist of four bases, namely adenine (A), thymine (T), guanine (G), cytosine (C).The governing principle in its duplication is the well known Watson-Crick property (WCC).According to WCC A and T bound to each other and G and C bound to each other on the opposite strands.A and G are called the complements of T and C respectively or vice versa.
Acting like an error-correcting code in nature, the DNA, has naturally attracted the attention of coding theorists in their researches.Consequently the concept of a DNA-code, that is a code that has special properties (i.e.reversible and complement) akin to the DNA, was introduced and has been a focal point of research in recent years.In many other studies, some constraints such as the Hamming distance constraint, the reverse constraint, the reverse-complement constraint and the fixed GC-content constraint are also considered [7,9,10,12].In [1], Aboluion et al. develop a new approach and extend the studies of [8] and [9].They added the reverse-complement constraint to further prevent the unwanted hybridizations.In such works as [2,[7][8][9], the focus was on constructing large sets of DNA codewords by using minimum Hamming distance.
With the four letter ambient alphabet of the DNA, many of the early works on DNA codes, use algebraic structures of size four to construct DNA codes.To this end, we can cite [2], in which DNA codes are studied over the Galois field of size 4.As another possible such alphabet, Siap et al. studied DNA codes over the finite ring F 2 [u]/(u 2 − 1) in [19].In [15], DNA-double pairs are used with the field F 16 and optimal codes are obtained.In [17], a generalization of [15] was done to include DNA 2k-bases (DNA pairs) over a suitable ring.
In this paper, we extend on the work done in [20], in which DNA pairs were matched with the elements of the ring F 2 [u]/(u 4 − 1) of size 16.Here, we match DNA 2 k−1 -bases (mers) {AA...AA . The main idea of the paper is to lay out the theory behind generating reversible and reversible-complement DNA codes over this ring.In doing so, we first tackle the issue of ring-reversibility versus the DNA reversibility.A problem that arises in all the cases where DNA k-mers (k ≥ 2) are matched with ring elements, it can be best explained in the following example: Let (u 1 , u 2 , u 3 ) ∈ R 4 be a codeword that corresponds to the DNA string ATAGCC.The reverse of (u 1 , u 2 , u 3 ) is (u 3 , u 2 , u 1 ) and (u 3 , u 2 , u 1 ) corresponds to CCAGAT.But CCAGAT is not the reverse of ATAGCC.This problem is solved by using the so-called u 2 -adic system.After giving the theoretical results concerning the constructions of DNA codes over the ring, we give examples of DNA-codes thus obtained.
The rest of the work is organized as follows.In section 2, we give some brief background on the ring R 2 k .In section 3, we introduce the u 2 -adic digit system for the ring R 2 k and demonstrate how it can be used to solve the reversibility problem.In section 4, we give the main results about cyclic DNA codes over R 2 k together with many examples.We finish the paper in section 5 with some concluding remarks and directions for possible future research on the related topics.

Preliminaries
We consider the finite chain ring R in this work.As can easily be observed from its structure, R 2 k is a commutative, characteristic 2 ring of size 2 2 k .The chain of ideals can be listed as A linear code over R 2 k of length n is defined naturally as an R 2 k -submodule of (R 2 k ) n .By using certain elementary row and column operations, it can be shown that any linear code C over R 2 k of length n is equivalent to a code whose generating matrix can be put into the following standard form: where M j i 's are matrices over R 2 k .As is the case with codes over finite chain rings, instead of the dimension, we can talk about the type of a code.A code with the above generator matrix is said to have type (t 0 , t 1 , t 2 , • • • , t 2 k −1 ) and the size of the code is given by In studying cyclic codes, it is essential to make use of their algebraic structures by matching each codeword . Thus cyclic codes of length n in that case correspond to the ideals of the quotient ring R 2 k [x]/(x n − 1).As will be needed later, we include in this section, the definition of the reciprocal polynomial.For A code C is said to be reversible if c r ∈ C for all c ∈ C, where c r corresponds to the reverse of c.
, where c c denotes a suitably defined complement of c.In general, algebraically the complement corresponds to adding a fixed constant to each of the coordinates.A code is said to be a reversible-complement code if it is both reversible and complement.DNA codes are defined as codes that satisfy both the reversible and complement property.
3. u 2 -adic digit system for DNA 2 k−1 -mers In [16], the u 2 -adic digit system was introduced for DNA k-mers.We will make use of the function η from [16], which operates on DNA single bases in the following form: The u 2 -adic digit system can be used for all the rings of the form R 2k = F 2 [u]/(u 2k −1), in particular for the ring R 2 k , that is in question in our work.
The following definitions are obtained from [16] by modifying them to go along with the ring R 2 k : Definition 3.1.Every element of the ring R 2 k can be expressed as a linear combination of 1, u 2 , u 4 , . . ., u 2 k −2 , with coefficients from {0, 1, u, 1 + u} = F 2 [u]/(u 2 − 1).We can view this as a numberbase system with the digits 1s (units), u 2 s, u 4 s, u 6 s, u 8 s, ... , u 2 k −2 s.We call this system the u 2 -adic system.
This corresponds to an element in the ring R 2 k , where this correspondence is given by the following: where The following lemma expresses the effect of multiplying by u 2 : Proof.According to the structure of u 2 -adic digits, multiplying by u 2 shifts the u 2 -adic digits cyclically to the left.The result then follows from the definition of ζ.
For p(u) ∈ R 2 k , let p(u) u 2 or p u 2 represent the u 2 -adic digits of p(u) as follows: Denote by ε, the function ε(p u 2 ) = p(u) and by [p u 2 ] r , the reverse vector, namely [p u 2 ] r = a 0 a 1 a 2 ...a (2 k−1 −1) .Then by Lemma 3.3 we have , Digits: The following theorem provides us with reverse-complement sets of DNA 2 k−1 -mers.
Theorem 3.5.Every ideal in R 2 k corresponds to a reversible-complement set of DNA 2 k−1 -mers.
Proof.We know, from the ideal structure of R 2 k , that every ideal is principally generated by (1 + u) m for some m = 0, 1, 2, . . . 2 k − 1.So let (1 + u) m be any ideal.We will consider two cases separately.
Case 1: m is even.
where 0 corresponds to a vector of zeros of length .We assume that c 1 = 0. Now, since Then, for 0 ≤ i ≤ we have (in u 2 -adic system) r , we see that u(1 + u) m is also self-reversible.But then this implies, by linearity of self-reversability, that any element in the ideal is self-reversible.Case 2: m is odd.Then we can write We can then use Case 1 for (1 + u) m−1 .If we then take any polynomial q(u) = since the reversibility is linear and closed under multiplying by a + ub and u 2 .This proves that every ideal corresponds to a reversible set of DNA 2 k−1 -mers.
According to the Watson-Crick complement and the correspondence we have defined the complement of an element α ∈ R 2 k is just given by α+1+u+u 2 +• • •+u 2 k −1 .Because of the ideal structure, we know that each ideal contains the generator of the minimal ideal, which is given by we see that the complement of each element in I is also in I.
This proves that each ideal corresponds to a reversible-complement set of DNA 2 k−1 -mers.
Let us give an illustrative example.
Take q(u) = u 6 + u + 1.And let us compute [(q(u)p(u)) u 2 ] r .First, we see that Notice that p(u) = u 2 + 1 and so, This means = 2. Thus we have Hence, Now let us verify that indeed we get the reverse DNA 4-mer:

Cyclic codes for DNA codes over R 2 k
Cyclic codes, have been one of the most common methods by which reversible-complement codes over different alphabets have been obtained.With the vast literature on the structural properties of cyclic codes over finite chain rings especially, this proves to be a fruitful direction to take.We will follow the footsteps of similar works to this effect.The following two theorems can easily be proven by following the similar steps as in [14] and [5].
Then the dual of the code is given by where fi = (x n − 1)/f i and if Further, if C is cyclic code of even length and f The following is a classical result about reversible codes: Theorem 4.2.[13] The cyclic code generated by a monic polynomial g(x) is reversible if and only if g(x) is self-reciprocal where g(x)|(x n − 1).
The following lemma, taken from [2] explains the basic properties of reciprocals.
We are now ready to state the main result of this section: Then the dual of the code is given by with fi = (x n − 1)/f i .Moreover, ) is a reversible DNA code.
• If f i 's ( fi 's) are self reciprocal polynomials and f 0 ( f * 2 k −1 ) divides (x n − 1)/(x + 1) over F 2 then C (C ⊥ ) is a reversible cyclic code with complement property and so , and hence by Lemma 4.3, we have Therefore, C is reversible.This means, by Theorem 3.5, that ζ −1 (C) is a reversible DNA code.
If moreover, f 0 divides (x n − 1)/(x + 1), then all f i 's divide (x n − 1)/(x + 1) since The proofs for C ⊥ , being similar to the above case, are omitted here.
By Theorem 4.4, C is a reversible cyclic code over R 2 3 and it corresponds to a reversible-complement DNA code of length 24.The dual is given by : Then all these polynomials are self reciprocal but not all of them divide (x n − 1)/(x + 1).Thus C ⊥ corresponds to a reversible DNA code.
By the same argument as in the previous example, C is a reversible cyclic code over R 2 3 and moreover, since the polynomials satisfy the complement properties, C corresponds to a reversible-complement DNA code of length 24.
The next example illustrates the DNA reversibility in an explicit form.
, where we will consider the codes over R 8 .Suppose C = ((1 + u) 3 f 0 , (1 + u) 4 f 1 ), whose generator matrix can be written as follows: Let We can easily observe that the reverse of v 2 is given by v r 2 = (1 + u)v 1 + v 2 + v 3 , while v 1 and v 3 are self-reversible.Consider the codeword whose DNA-correspondence is given by G, T, G, A, A, A, A, A, A, A, A, T, G, T, G, A, A, A, A, A, A, A, A, T, G, T, G,  A, A, A, A, A, A, A, A).

Note that
ζ −1 (c 2 ) =(A, A, A, A, A, A, A, A, G, T, G, T, A, A, A, A, A, A, A, A, G, T, G, T, A, A, A, A, A, A, A, A, G, T, G, T ) and we have Next, consider e 1 = uv 1 + (1 + u 3 )v 2 + (u 2 + u 3 )v 3 , which is given as a vector by so that the DNA correspondence is given by Now, if we let e 2 = ((u , which then would be written as a vector as we will have T, G, A, C, A, G, T, T, G, A, C, C, T, G, A, C, A, G, T, T, G, A, C, C, T, G, A, C, A, G, T, T, G, A, C).
Example 4.8.Let us consider self reciprocal polynomials f 0 = x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x Then, C is a reversible cyclic code over R 2 4 and it corresponds to a reversible-complement DNA code of length 120 and minimum Hamming distance 2.
Example 4.9.Consider f 0 = x 3 + x 2 + x + 1 and In Table 1 we give a table of some reversible-complement codes obtained from similar constructions.
We conclude this section by the following theoretical result whose proof is omitted, being similar to the above ones, with an application.2: Table 2. f = (x − 1) and g = (x 2 + x + 1)

Ring Length
Generator of the code

Conclusion
In this work, we solve the reversibility problem on cyclic codes with ideal structures of the ring R 2 k by u 2 -adic system.Both reversible cyclic codes on the ring R 2 k and reversible (or reversible complement) DNA codes are obtained from the same codes.The properties of DNA codes with the family of cyclic dual and self dual codes have been explored.Thus an extension of what was done in [20] has been established in the most general sense without puncturing.The idea that we have used can be used for different such extensions as well.However, the finite chain property of the ring, which is useful in both describing all the ideals of the ring and making it easier to study cyclic codes, has been essential in obtaining our main results.When the extension ring does not have the finite chain property, some partial results can still be obtained, however a complete characterization, similar to what we have done here, is very challenging.Hence, future studies on the general case i.e reversible and complement cyclic codes over different rings such as R 2k remains to be an open and interesting problem.

Example 4 . 5 .
Let us consider the self reciprocal polynomials f 0

Example 4 . 11 .
Some self dual codes that correspond to reversible complement DNA codes are shown in Table

Table 1 .
f = 1 + x, g = 1 + x + x 2 .Reversible cyclic codes over R 2 3 of length 6 which correspond to reversible complement DNA codes of length 24.(d(C): Minimum Hamming distance of the code C. d(ζ −1 (C)): Minimum Hamming distance of the code ζ 7f 1 ) be the code over R 23.Then C is a cyclic code of length 4 and distance 2. ζ −1 (C) is a reversible-complement DNA code of length 16 and Hamming distance 4.