Skew Cyclic Codes Over $\mathbb{F}_4 R$
Nasreddine Benbelkacem Martianus Frederic Ezerman Taher Abualrub Aicha Batoul

Combining Non-probability and Probability Survey Samples Through Mass Imputation
    Jae Kwang Kim^1,
  	 Seho Park^2, Yilin Chen^3, and Changbao Wu^3

    ^1 Department of Statistics, Iowa State University,

     ^2 Department of Biostatistics, Indiana University School of Medicine, 

    ^3 Department of Statistics and Actuarial Science,
University of Waterloo
    
===========================================================================================================================================================================================================================================================================================




Nasreddine Benbelkacem^*
Faculty of Mathematics, University of Science and Technology Houari Boumediene,
BP 32 El Alia, Bab Ezzouar, 16111 Algiers, Algeria




Martianus Frederic Ezerman
School of Physical and Mathematical Sciences, Nanyang Technological University,
21 Nanyang Link, Singapore 637371




Taher Abualrub
Department of Mathematics and Statistics, College of Arts and Sciences,
	American University of Sharjah, P.O. Box 26666 Sharjah, United Arab Emirates



Aicha Batoul
Faculty of Mathematics, University of Science and Technology Houari Boumediene,
	BP 32 El Alia, Bab Ezzouar, 16111 Algiers, Algeria





 (Communicated by the associate editor name)



This paper considers a new alphabet set, which is a ring that we call _4R, to construct linear error-control codes. Skew cyclic codes over the ring are then investigated in details. We define a nondegenerate inner product and provide a criteria to test for self-orthogonality. Results on the algebraic structures lead us to characterize _4R-skew cyclic codes. Interesting connections between the image of such codes under the Gray map to linear cyclic and skew-cyclic codes over _4 are shown. These allow us to learn about the relative dimension and distance profile of the resulting codes. Our setup provides a natural connection to DNA codes where additional biomolecular constraints must be incorporated into the design. We present a characterization of R-skew cyclic codes which are reversible complement.




§ INTRODUCTION


The use of noncommutative rings to construct error control codes has recently been an active research area, initiated by the seminal works of Boucher in <cit.> and <cit.> as well as that of Abualrub in <cit.>. In the first two, Boucher and collaborators generalized the notion of cyclic codes by using generator polynomials in a noncommutative polynomial ring called the skew polynomial
ring. They supplied examples of skew cyclic codes with Hamming distances larger
than previously best-known linear codes of the same length and dimension.  In <cit.>, Abualrub generalized the concept of skew cyclic codes to skew quasi-cyclic codes. They then constructed several new codes with Hamming distances exceeding the Hamming distances of the previously best-known linear codes with comparable parameters. 

Another emerging topic in the studies of error correcting codes is additive codes over mixed alphabets. Borges introduced the class of _2_4-additive codes in <cit.>. This class generalizes binary and quaternary linear codes. A _2_4-additive code is defined to be a subgroup of _2^r_4^s. Abualrub studied the structure of _2_4-additive cyclic codes in <cit.>. They determined the generator polynomials and the minimal generating sets for these codes.

In this paper, we merge the topic of skew cyclic codes with that of codes over mixed alphabets. In particular, we study the structure of linear skew cyclic codes over the ring _4R where _4 is the finite field of four elements and R={a + v b | a,b ∈_4} is the commutative ring with 16 elements with v^2=v. Any codeword  in a skew cyclic code C over _4R has the form =(a_0,a_1,...,a_α -1,b_0,b_1,...,b_β -1)∈_4^αR^β. 

The followings are our contributions. 

	
  * We show that the dual of a skew cyclic code over _4R is also a skew cyclic code. In fact, skew cyclic codes over _4R are left R[X,θ ]-submodules of 
	R_α ,β=_4[X]/⟨ X^α-1⟩× R[X,θ ]/⟨ X^β-1⟩.
	 
	
  * We determine their generator polynomials and establish interesting results that relate these codes to cyclic and quasi-cyclic codes over _4 R. First, we show that a skew cyclic code over _4R is equivalent to an _4R-cyclic code if α and β are both odd integers. Second, we establish that if α and β are both even integers, then an _4R-skew cyclic code C is equivalent to an _4R quasi-cyclic code of index 2. 
	
	
  * Conditions for skew cyclic codes over _4R to be self-orthogonal are studied.
	
	
  * We use the Gray mapping to associate these codes to codes over _4 of length α +2 β and exhibit a nice relationship between these codes and their images over _4. The Gray image of any skew cyclic code over _4R is the product of a cyclic code over _4 of length α and two skew cyclic codes, each of length β. We supply examples of skew cyclic codes over _4R and their respective Gray images for different lengths. 
	
	
  * Applications of these codes to DNA computing are included in our treatment.





§ PRELIMINARIES

	
Let _4={0,1,w,w^2=w+1} and R={a+ vb | a,b in _4} be, 
respectively, the finite field with four elements and the commutative ring with 16 elements where v^2=v. It is well-known that R is a finite non-chain ring with two maximal ideals ⟨ v⟩ and ⟨ v+1⟩, making R/⟨ v⟩ and R/⟨ v+1⟩ isomorphic to _4. The Chinese Remainder Theorem then implies that R=⟨ v⟩×⟨ v+1⟩. As was shown in <cit.>, R can be uniquely expressed as {a + vb = (b+a) v + a (v+1) | a,b∈_4}.
	
Let A⊕ B={a+b| a∈ A,b∈ B} and A⊗ B={( a,b) | a∈ A,b∈ B} as defined in <cit.>. An _4-linear code of length n is a subspace of _4^n. A subset C of R^n is a linear code over R if C is an R-submodule. Given a linear
code C over R, let 

    C_1   ≜{𝐱 + 𝐲∈_4^n| (𝐱+ 
    𝐲) v + 𝐱 (v+1) ∈ C 𝐱, 
    𝐲∈_4^n}
    
    C_2   ≜{𝐱∈_4^n| (𝐱 + 𝐲) v + 
    𝐱 (v+1) ∈ C 𝐲∈_4^n}.

One can quickly verify that C_1 and C_2 are linear codes over _4. In fact, any linear code C over R can be expressed as C=v C_1⊕ (v+1) C_2. Let r = a + v b ∈ R and =(c_1,c_2,…,c_n) ∈ C, i.e., c_j=a_j + v b_j with a_j,b_j ∈_4 for 1 ≤ j ≤ n. The j-th entry of r is 

    (a + v b) (a_j + v b_j)    = ((a+b) v + a (v+1))(a_j + v b_j) 
       = a a_j_x + v (a b_j + b a_j + b b_j_y) =
    (x+y) v + x (v+1).

Hence, r can be written in terms of C_1 and C_2 with 

    𝐱 = a (a_1, a_2, …, a_n) 𝐲 = (a+b)(b_1,b_2, …, b_n) + b (a_1, a_2, …, a_n).



Let an automorphism θ over R be defined by

    θ : R ↦ R  a+vb ↦ a^2+(v+1) b^2.

Restricted to _4, it interchanges w and w^2 while keeping {0,1} fixed. Note that our θ here is equal to the composition of automorphisms φ∘θ_1 in <cit.>. A subset C of R^n is said to be an R-skew cyclic code of length n if two conditions are satisfied.

	
  * C is an R-submodule of R^n.
	
  * If = (c_0,c_1,...,c_n-1)∈ C then the skew cyclic shift of  over R, denoted by T_θ() 
	≜ (θ(c_n-1),θ (c_0),...,θ (c_n-2)), must also be in C.



It is often convenient to associate a vector 𝐚 = (a_0,a_1,…,a_n-1) with a polynomial a(X):=a_0+a_1 X + … + a_n-1 X^n-1 in an indeterminate X. This allows for constructions of codes using results from the algebra of polynomial rings.
	
The next two theorems can be inferred by a slight modification on the
corresponding theorems in <cit.> with q restricted to 4. The respective proof is therefore omitted for brevity.
	
(From <cit.>) 
Let C=v C_1⊕ (v+1) C_2 be a linear code over R. Then C is an R-skew cyclic code if and only if C_1 and C_2 are skew cyclic codes over _4.

	
(From <cit.>) 
Let C=v C_1⊕ (v+1) C_2 be a skew cyclic code of
length n over R. Let g_1(X) and g_2(X) be the respective generator polynomials of C_1 and C_2 as _4-skew cyclic codes. Then C=⟨ v g_1(X) + 
(v+1) g_2(X) ⟩.

	
For any element in R, we introduce a new ring homomorphism 

    η : R ↦_4 a+vb  a.

Let _4R := {(a,b) | a ∈_4 b∈
R}. It is straightforward to verify that _4R is an R-module under the multiplication 

    d ∗ (a,b)=(η ( d) a,db)  d ∈ R  (a,b)∈_4R.

This extends naturally to _4^α R^β. Let 𝐱=(a_0,a_1,...,a_α -1,b_0,b_1,...,b_β-1)∈_4^αR^β, for α and β∈ℕ, and d∈ R. Then 

    d ∗𝐱 = (η ( d) a_0,η ( d) a_1,...,η ( d) a_α-1,d b_0,d b_1,...,d b_β -1).

	

A nonempty subset C of _4^αR^β is called an 
_4R-linear code if it is an R-submodule of _4^αR^β with respect to the scalar multiplication in Equation (<ref>).

A nondegenerate inner product between 𝐱=(a_0,a_1,…,a_α-1, b_0,b_1,…,b_β-1) and 
𝐲=(d_0,d_1,…,d_α-1, e_0,e_1,...,e_β -1)
is given by 

    ⟨𝐱, 𝐲⟩ =v∑_i=0^α-1a_id_i+∑_j=0^β -1b_je_j∈ R.

The dual code of an _4 R-linear code C, denoted by C^⊥, is also _4R-linear and is defined in the usual way as 

    C^⊥:={𝐲∈_4^αR^β|⟨𝐱, 𝐲⟩ = 0 𝐱∈ C}.

	
Let 

    a(X)     =a_0 + a_1 X + … + a_α -1 X^α -1∈_4[X]/⟨ X^α-1⟩
    
    b(X)     = b_0 + b_1 X + … + b_β-1 X^β -1∈
    R[X,θ]/⟨ X^β-1⟩.

Then any codeword =(a_0,a_1,...,a_α-1, b_0,b_1,...,b_β -1)∈_4^αR^β can be
identified with a module element consisting of two polynomials such that 

    c(X)=(a(X),b(X)).

This identification gives a one-to-one correspondence between _4^αR^β and 

    R_α ,β≜_4[X]/⟨ X^α-1⟩× R[X,θ ]/⟨ X^β-1⟩.

Let r(X)=r_0 + r_1 X+ … + r_t X^t∈ R[X,θ] and (a(X),b(X)) ∈ R_α ,β. Their product is 

    r(X) ∗ (a(X),b(X))=(η (r(X)) a(X) , r(X)b(X))

where η(r(X))= η ( r_0 ) + η ( r_1 ) X + … + η (r_t)
X^t∈_4[X]. Here, η(r(X))a(X) is the usual polynomial
multiplication in _4[X]/⟨ X^α-1⟩ while r(X) b(X) is the polynomial multiplication in R[X,θ ]/⟨ X^β-1⟩ where X (a + v b) = ( a^2 + (v+1) b^2 ) X.
	

R_α,β is a left R[X,θ ]-module with respect to ∗ in Equation (<ref>).

	

Verifying that the required properties are satisfied over _4[X]/⟨ X^α-1⟩ is easy since we do not have to deal with skewness. Verifying over R[X,θ ]/⟨ X^β-1⟩ is routine, albeit tedious. It suffices to use the facts that θ is a homomorphism with θ^-1=θ.




§ GENERATOR POLYNOMIALS OF _4 R-SKEW CYCLIC CODES

	
This section begins with a formal definition of an _4R-skew cyclic code and proposes a method to determine the generator polynomial of any _4R-skew cyclic code C in R_α,β. We say that two codes are equivalent if one can be obtained from the other by some composition of a permutation of the first α positions, a permutation of the last β positions, and multiplication of the symbols appearing in a chosen position by a nonzero scalar.


An _4 R-linear code C of length n=α+β is said to be _4R-skew cyclic if, for any codeword =(a_0,a_1,…,a_α -1,b_0,b_1, …, b_β -1)∈ C, its skew cyclic shift 
T_θ() ≜ (a_α -1,a_0,…,a_α -2,θ
(b_β -1),θ (b_0),…,θ (b_β -2)) is also in C.

	

Let C be an _4R-skew cyclic code of length n=α +β
such that β is an even integer. Then C^⊥ is also an _4R-skew cyclic code of the same length.

	

It suffices to show that, for any  
=(a_0,a_1,…,a_α -1,b_0,b_1,…,b_β -1)∈ C^⊥, we have T_θ()∈ C^⊥. Let   
=(d_0,d_1,…,d_α -1,e_0,e_1,…,e_β -1) be any codeword in C. Then  

    T_θ () = 
    ⟨ ( a_α -1,a_0,…,a_α -2,θ (b_β-1),θ (b_0),…,θ (b_β -2)),(d_0,d_1,…,d_α -1,e_0,e_1, … , e_β -1)⟩
    
     =v(a_α -1d_0+a_0d_1+…+a_α -2d_α -1)+(θ(b_β-1)e_0+θ (b_0)e_1+…+
    θ (b_β -2)e_β-1).

Hence, one only needs to show that 

    0 = a_α -1d_0+a_0d_1+…+a_α-2d_α -1 
    0 = θ (b_β -1)e_0+θ
    (b_0)e_1+…+θ (b_β -2)e_β -1.

Now, let γ :=(α ,β ). Then γ is an even integer since β is an even integer. Since C is _4R-skew cyclic, for any ∈ C we have T_θ^γ() = and T_θ^γ -1() ∈ C. Hence, T_θ^γ -1 () =0. 
Since T_θ^γ -1 () = ( d_1,…,d_α -1,d_0,θ (e_1),…,θ(e_β -1),θ (e_0)), we then obtain 

    v ∑_j=0^α-1 a_j   
    d_(j+1) α + ∑_j=0^β-1 b_j  θ(e_(j+1) β)=0.
 
This implies

    0     = a_α -1d_0+a_0d_1+a_1d_2+…+a_α-2d_α -1
     
    0     = b_β -1θ (e_0)+b_0θ (e_1)+b_1θ(e_2)+…+b_β -2θ (e_β -1).

Applying θ to both sides of the last equation yields

    θ(0)=θ (b_β -1)e_0+θ (b_0)e_1+θ(b_1)e_2+…+θ (b_β -2)e_β -1=0,

completing the proof. 

	

A code C is _4R-skew cyclic if and only if C is a left 
R[X,θ]-submodule of R_α ,β under the multiplication ∗.

	

Let c(X)=(a(X),b(X)) be any codeword of an _4R-skew cyclic code C. Hence, (a_0,a_1,…,a_α-1,b_0,b_1,…,b_β-1) and all of it's T_θ-skew cyclic shifts are in C. We associate, for each j ∈, the polynomial

    X^j∗ c(X) =(a_α-j + a_α-j+1 X + … +
    a_α-j-1 X^α -1, 
    θ^j (b_β-j) + θ^j (b_β-j + 1) X + … + θ^j (b_β-j-1) X^β-1)

with the vector 

    (a_α-j,a_α-j+1,…,a_α-j-1, 
    θ^j(b_β-j),θ^j(b_β-j+1),…,θ^j(b_β-j-1)).

The indices of the first block (of length α) are taken modulo α and those of the second block (of length β) are taken modulo β. By the _4 R-linearity of C, we have r(X) ∗ c(X) ∈ C for any r(X) ∈ R[X,θ]. Thus, C is a left R[X,θ ]-submodule of R_α,β.
		
Conversely, let C be a left R[X,θ]-submodule of the left R[X,θ]-module R_α,β. Then, for any c(X) ∈ C, we have X^j ∗ c(X) ∈ C for any j ∈. Thus, C is indeed an _4R-skew cyclic code. 

	
Let C be an _4R-skew cyclic code. Let c(X)=(a(X),b(X)) be an element in C. Let ℓ(X) be an element in _4[X]/ ⟨ X^α-1⟩. We use 0 to denote either the zero vector (0,0,…,0) or the zero polynomial. Let 

    I    ≜{b(X)∈ R[X,θ] /⟨ X^β-1⟩|
    (ℓ(X),b(X))∈ C }
    
    J    ≜{a(X)∈_4[X]/⟨ X^α-1⟩|
    (a(X),0) ∈ C}.

The next results establish useful properties of the sets I and J
	

J is an ideal in _4[X]/⟨ X^α-1 ⟩ generated by a left divisor of X^α-1.

	

Let a_1(X) and a_2(X) be elements of J. By definition, (a_1(X),) and (a_2(X),) are in C. Hence, (a_1(X),) + (a_2(X) ,) = (a_1(X) + a_2(X),) ∈ C, making a_1(X) +a_2(X) ∈ J. Let s(X) ∈_4[X]/⟨ X^α-1 ⟩ and a(X) ∈ J. Then (a(X),) is in C. Because C is a left R[X,θ]-module, we have 

    s(X) ∗ (a(X),)=(s(X) a(X),) ∈ C  s(X) a(X)  (X^α-1) ∈ J.

Thus, J is an ideal in _4[X]/⟨ X^α-1 ⟩ generated by a left divisor f(X) of X^α-1.

	

I is a principally generated left R[X,θ]-submodule of R[X,θ]/⟨ X^β-1 ⟩.

	

Let b_1(X) and b_2(X) be elements in I. Then there exist polynomials  ℓ_1(X) and ℓ_2(X) in _4[X]/⟨ X^α-1 ⟩ such that  (ℓ_1(X),b_1(X)),(ℓ_2(X),b_2(X)) ∈ C. Hence, 

    (ℓ_1(X),b_1(X)) +(ℓ_2(X),b_2(X))=
    (ℓ_1(X) +ℓ_2(X) ,b_1(X) +b_2(X)) ∈ C,

implying b_1(X) +b_2(X) ∈ I. Let r(X)∈ R[X,θ]/⟨ X^β-1 ⟩ and (ℓ(X), b(X))∈ C. Since C is a left R[X,θ]-submodule of R_α,  β, we have  

    r(X) ∗ (ℓ(X),b(X)) = (η(r(X)) ℓ(X)  (X^α-1), r(X) b(X)  (X^β-1))

in C, making 
r(X) b(X)  (X^β-1) ∈ I. Thus, I is a left submodule in 
R[X,θ]/ ⟨ X^β-1 ⟩ and, by Theorem <ref>, 
I=⟨ g(X) ⟩ where 

    g(X) ≜ v g_1 (X) + (v+1) g_2(X).


	
The following result classifies all _4 R-skew cyclic codes.

Let g(X) be as defined in Equation (<ref>). Let C be an _4R-skew cyclic code. Then C is generated as a left submodule of R_α,β by (f(X),) and (ℓ(X),g(X)) where ℓ(X) is an element in _4[X]/⟨ X^α-1 ⟩ and f(X) is a left divisor of X^α-1.

	

Let =(_1,_2) ∈ C with _1 ∈_4^α and _2 ∈ R^β. Then c_2(X) ∈ I and we write c_2(X) = q(X)   g(X) for some q(X) ∈ R[X,θ]/⟨ X^β-1⟩. There exist ℓ(X) ∈_4[X]/⟨ X^α-1⟩ such that  (ℓ(X),g(X)) ∈ C since g(X) ∈ I. We have 

    =(_1,_2)    = (c_1(X),) + (, q(X) g(X)) 
       = (c_1(X),) + (η(q(X)) ℓ(X), q(X) g(X)) + (η(q(X)) ℓ(X), )
       = (c_1(X),) + q(X) ∗((ℓ(X),g(X)) + (ℓ(X), )).

Hence, (η(q(X)) ℓ(X) + c_1(X), ) ∈ C, making η(q(X)) ℓ(X) + c_1(X) ∈ J. By Lemma <ref>, there exists p(X) ∈ J satisfying η(q(X)) ℓ(X) + c_1(X) = p(X) f(X). Thus, c(X) = q(X) ∗ (ℓ(X), g(X)) + (p(X) f(X),).

	

Let C be an _4R-skew cyclic code. Then, 
without loss of generality, we can assume (ℓ(X)) <  (f(X)).

	

Suppose that (ℓ(X)) - (f(X)) = k ≥ 0. Consider the code D generated by the set

    {(f(X),),(ℓ(X),g(X)) + s X^k∗ (f(X),)} =
    {(f(X),),(ℓ_1(X),g(X))}

where ℓ_1(X) = ℓ(X)+s X^k f(X) for some s ∈_4. Hence, D ⊆ C. On the other hand, 

    (ℓ(X),g(X)) =(ℓ(X),g(X)) +s X^k∗ (f(X),) - s X^k∗ (f(X),).

Hence, C⊆ D, making C=D. Notice here that (ℓ_1(X)) < (ℓ(X)). We repeat the same process on ℓ_1(X) until we obtain (ℓ(X)) < (f(X)). 

	

An _4R-skew cyclic code is equivalent to an 
_4R-cyclic code if both α and β are odd integers.

	

Let C be an _4R-skew cyclic code and 
γ :=(α,β). Then  (γ,2)=1 since γ is odd. Then there exist integers k and j such that γ k + 2 j=1 and, hence, 2 j = 1 - γ k = 1 + γ t for some t > 0 where t ≡ -k γ. As in Equation (<ref>), let 
c(X)=(a(X), b(X)) ∈ C. Then 

    X^2 j∗ c(X)    = X^2 j∗(∑_i=0^α -1 a_i X^i  ,
    ∑_i=0^β -1b_iX^i ) =
    (∑_i=0^α-1 a_i X^i+2 j , 
    ∑_i=0^β-1θ^2 j (b_i) X^i+2 j)
       = (∑_i=0^α-1 a_i X^i+1+ γ t, 
    ∑_i=0^β-1θ^2 j (b_i) X^i+1+ γ t)
        = (∑_i=0^α-2 a_i X^i+1+γ t + 
    a_α-1 X^α +γ t , 
    ∑_i=0^β-2 b_i X^i+1+γ t
    + a_β-1 X^β +γ t) 
       = (∑_i=0^α-2 a_i X^i+1 + a_α-1 ,
    ∑_i=0^β-2 b_i X^i+1 + b_β-1).

The second to the last equation is due to θ^2(r)=r for all r ∈ R while the last equation follows because X^α =X^β=X^γ=1.

	

An _4R-skew cyclic code is equivalent to an 
_4R-quasi-cyclic code of index 2 if both α and β are even integers.

	

Let C be an _4R-skew cyclic code, α=2N, and β=2M for some N,M ∈. Then γ =(α,β) is an even integer with (γ,2)=2. 
For any 

    =(a_0,0, a_0,1, …, a_N-1,0, a_N-1,1 , b_0,0, b_0,1, …, b_M-1,0, b_M-1,1) ∈ C

there exist integers k ≥ 0 and j such that 2 j = 2 + k γ. Consider 

    T_θ^2+ k γ( a_0,0,a_0,1,…,a_N-1,0,a_N-1,1 , 
    b_0,0,b_0,1,…,b_M-1,0,b_M-1,1) =  
       T_θ^k γ(a_N-1,0,a_N-1,1,…,a_N-2,0,a_N-2,1 ,
    b_M-1,0,b_M-1,1,…,b_M-2,0,b_M-2,1) = 
       ( a_N-1,0,a_N-1,1,…,a_N-2,0,a_N-2,1,
    b_M-1,0,b_M-1,1,…,b_M-2,0,b_M-2,1) ∈ C

since T_θ^k γ() = for any ∈_4^αR^β. 
Thus, C is equivalent to an _4R-quasi cyclic code of length n=α +β and index 2.




§ THE GRAY MAPPING


The classical Gray mapping ϕ^∗: R ↦_4^2 is defined by ϕ^∗(a+vb)=(a+b,a) for any a + v b ∈ R. The Lee weight of any element in R is the Hamming weight of its image under ϕ^∗. This map extends naturally to vectors in R^n. For any 𝐱=(x_0,x_1,…,x_α-1) ∈_4^α and 
𝐲=(y_0,y_1,…,y_β-1) ∈ R^β, the Gray map over _4 R is defined by 

    ϕ : _4^αR^β↦_4^α+ 2
    β (𝐱,𝐲)  (𝐱,ϕ^∗(𝐲)).

The map ϕ is an isometry which transforms the Lee distance in _4^αR^β to the Hamming distance in _4^α + 2 β. For any _4R-linear code 
𝒞, the code ϕ(𝒞) is _4-linear. Furthermore, we have 

    wt(𝐱,𝐲)=wt_H(𝐱)+wt_L(𝐲)

where wt_H(𝐱) is the Hamming weight of 𝐱 and 
wt_L(𝐲) is the Lee weight of 𝐲.
	

Let C be a self-orthogonal _4R-linear code under the inner product defined in Equation (<ref>). Then ϕ(C) is a Euclidean self-orthogonal code over _4.

	

It suffices to show that the Gray images of codewords are Euclidean orthogonal
whenever the codewords are orthogonal. Let 𝒞 be a self-orthogonal _4R-linear code of length α+β. Let =(, + v  ),  =(, + v  ) ∈_4^α× R^β be codewords in C with , ∈_4^α and 
, , , ∈_4^β. Then, by Equation (<ref>),

    ⟨, ⟩  
    = v (·) + · + v (· + · + ·)= 0 + v 0 ∈ R.

Hence, · =0 and · + · + 
· + · =0 .
Since ϕ() =(,  +  ,) and 
ϕ() =(,  +  , ), one gets   

    ϕ() ·ϕ() =
    · + · + · + 
    · + · + · =0.

Therefore, the code ϕ(C) is Euclidean self-orthogonal.



Let C be an _4 R-skew cyclic code of length n=α +2 β. Then, 
ϕ(C)=C_0⊗ C_1⊗ C_2, where C_0 is a cyclic code of 
length α in _4[X] /⟨ X^α-1⟩ and 
both C_1 and C_2 are skew cyclic codes of length β in 
R[X]/⟨ X^β-1⟩. Moreover, 
ϕ(C) = ∏_i=0^2C_i.



From {=(a_0,a_1,…,a_α-1,b_0 + v c_0, b_1 + v c_1,…, 
b_β-1 + v c_β-1) : ∈ C }, we construct the codes

    C_0 := {(a_0,a_1,…,a_α-1) } , 
    C_1 := {(b_0 + c_0, b_1 + c_1,…, b_β-1 + c_β -1) }, 
     C_2 := {(b_0, b_1,…, b_β -1)}.

A codeword :=(a_0,a_1,…,a_α -1) ∈ C_0 corresponds to a codeword 

    =(a_0,a_1,…,a_α-1,b_0 + v c_0, b_1 + v c_1,…, b_β-1 + v c_β-1) ∈ C.

Since C is an _4 R-skew cyclic code, we know that T_θ () is given by

    (a_α-1, a_0, a_1, …,a_α-2, θ(b_β-1 + v c_β-1), 
    θ(b_0 + v c_0),…,θ(b_β-2 + v c_β-2)) ∈ C.

Hence, ( a_α -1,a_0,a_1,… ,a_α -2) ∈ C_0. This implies that C_0 is a cyclic code of length α in _4[X]/⟨ X^α-1 ⟩. 

The proof that both C_1 and C_2 are skew cyclic codes of length β in 
R[X]/⟨ X^β-1⟩ follows the same line of argument. Thus, ϕ(C) = C_0⊗ C_1⊗ C_2 and ϕ(C) = 
∏_i=0^2C_i.     



Let C=⟨(f( x),),(,g( x))⟩ be an _4 R-skew 
cyclic code with ℓ(X):=. Then C = C_1⊗ C_2 where C_1 
is a skew cyclic code over _4 and C_2 is a skew cyclic code over R.

	

Note that =(_1, _2) ∈ C if and only if _1 =q_1 f(X) and 
_2=q_2 g(X) if and only if _1∈ C_1 = 
(f(X)) and _2∈ C_2=(g(X)) if and only if C=C_1⊗ C_2 
where C_1=(f(X)) and _2∈ C_2=(g(X)).

	
 
Let C= C_1⊗ C_2 where C_1 is an _4-skew cyclic Euclidean self-orthogonal code 
and C_2 is an R-skew cyclic self-orthogonal code over R. Then C is a self-orthogonal 
_4 R-skew cyclic code.

	

Suppose C_1⊆ C_1^⊥ and C_2⊆ C_2^⊥. 
Let =(_1, _2) ∈ C and = (_3,_4) ∈ C. 
Then _1,_3∈ C_1= (f(X)) and 
_2,_4∈ C_2=(g(X)). This implies that _1·_3 = 0 ∈_4 
and ⟨_2,_4⟩ = 0 ∈ R. 
Hence,

    ⟨,⟩ = v (_1·_3)
    +⟨_2,_4⟩ =v · 0 + 0 = 0.

Therefore, C⊆ C^⊥, implying that the _4R-skew cyclic code C is self-orthogonal.  

	
Note that the converse does not hold. In fact, C^⊥≠ C_1^⊥⊗ C_2^⊥ in general.
	


§ _4-CODES FROM _4R-SKEW CYCLIC CODES



	
The image of a code over a given ring under the Gray map is a code over a field. This latter code is usually inferior in terms of the usual measure of relative rate and relative distance when compared with linear codes directly constructed algebraically over the corresponding field. In this respect, _4 R-skew cyclic codes are not exempted. Their excellent structures, particularly as revealed in Theorem <ref>, allow us to determine good choices of the ingredient codes C_0, C_1, and C_2 that result in best-possible dimension and distance profiles. 

Since factorization in the skew polynomial ring _4[X,θ] requires considerably more care than in _4[X], we started by finding good codes based on the factorization of X^β-1 ∈_4[X,θ]. A search for good skew cyclic codes was done following the suggestion of Caruso and Le Borgne in <cit.>. Through personal communication Le Borgne sent us an implementation routine in MAGMA <cit.>. The identified good skew cyclic codes were subsequently used as C_1 or C_2.

Based on the particular structure described in Theorem <ref>, the three ingredient codes C_0,C_1, and C_2 are combined by the DirectSum routine in MAGMA to yield the code ϕ(C). To minimize the drop in the relative dimension and relative distance of ϕ(C), we chose the ingredient codes to have equal minimum distances and relatively small dimensions. Examples of good choices for the codes C_1 and C_2 are given in Table <ref> while Table <ref> contains the cyclic codes that we used as C_0. The best parameters of the resulting ϕ(C) are listed in Table <ref>.




	


§ DNA SKEW CYCLIC CODE OVER _4R

	
The encoding and decoding systems to store or transfer information or data
by mimicking DNA sequences are known collectively as DNA codes. 
The strands, i.e., DNA strings, are preferred to be short to
make the synthesis easy and cheap. They must, however, satisfy
numerous constraints to be useful for applications. The two most common
applications are as basic tools for biomolecular computation and as
biomolecular barcoding-tagging system to identify and manipulate individual
molecules in complex libraries.
	
Numerous approaches to DNA codes have been extensively investigated. A recent addition to several surveys that have appeared in the literature is	the work of Limbachiya in <cit.>. Tools from algebraic coding theory, both from finite fields as well as rings, have been fruitfully used since the inception. A relatively early work by Marathe in <cit.> discussed important design criteria and bounds derived from error-correcting codes. We continue on this line of studies by constructing _4R-DNA skew cyclic codes.
	
The Watson-Crick complement of a strand is the strand obtained by
replacing each 𝙰 by 𝚃 and vice versa, and each 𝙶 by 𝙲 and vice versa. One writes 𝙰
= 𝚃, 𝚃 = 𝙰, 𝙲
=𝙶, and 𝙶=𝙲. Let 𝐱
=(x_1,x_2,…,x_n) and 𝐲=(y_1,y_2,…,y_n) be distinct
codewords in a DNA code 𝒟. The reverse of 𝐱
is 𝐱_=(x_n,x_n-1, …, x_1). The 
complement of 𝐱 is 𝐱^c=(x_1,x_2,…, x_n). Hence, 𝐱_^c= (x_n,x_n-1, …,x_1) is the reverse complement of 𝐱.
	
The process in which a strand and its complement bound to form a double-helix is known as hybridization. Constraints on the codewords in a DNA code are imposed to avoid it. Let 𝒟 be a DNA code of fixed length n, cardinality M, and minimum distance d. Then the constraints on the Hamming distances 

    wt_H(𝐱,𝐲) ≥ d wt_H(𝐱^c,𝐲_) ≥ d 𝐱,𝐲∈𝒟

are imposed to prevent hybridization between any two strands as well as between a strand and the reverse of any other strand. A reverse-complement DNA code 𝒟 has parameters (n,M,d)_4 that satisfies Equation (<ref>). It is known, for instance, that any _4-cyclic code with generator polynomial f(X) is reverse-complement if and only if f(X) ∈_4[X] is a self-reciprocal polynomial, , f(X)=X^(f(X)) f(X^-1), not divisible by X-1.
	
Abualrub studied _4-DNA codes of odd lengths in <cit.> where they use the bijection between the set of DNA alphabets {𝙰,𝚃,𝙲,𝙶} and _4:={0,1,w,w^2}, in that respective ordering. We extend this idea by letting, for all a∈ R, 

    θ(a) + θ (a) = v + 1.

	

For all a,b ∈ R, we have

	
  * a+b=a+b+v
	
  * (v+1)a=(v+1)a+v.
	
  * va=va.


	
The map defines the following bijection between the elements of R and the 16 codons in {𝙰,𝙲,𝙶,𝚃}^2.
	

a ∈ R     Codon     a ∈ R     Codon     a ∈ R     Codon     a ∈ R     Codon

 0          v+w          vw^2+1          vw     
 
 
v          w          vw^2+w^2          vw+w^2
     
 
v+1          v+w^2          vw^2          vw+w
     
 
1          w^2          vw^2+w          vw+1     
 
 


An R-linear code C of length β is called DNA-skew cyclic if

	
  * The code C is R-skew cyclic of length β. 
	
  * For any codeword ∈ C, ≠_^c with _^c∈ C.


	
We adopt the following definition of reciprocal polynomials and a useful lemma from <cit.> .
	

Let f(X)=f_0 +f_1 X + … + f_k X^k be a polynomial in R[X,θ]. The reciprocal polynomial of f(X) is the polynomial f^⋆(X) given by

    f^⋆(X)  
    
    = f_0 X^k + f_1 X^k-1 + f_2 X^k-2 + … + f_k-1 X+ f_k.

If f(X)=f(X)^⋆, then f(X) is self-reciprocal.


 <cit.>
Let f(X),g(X) ∈ R[X,θ] with (f(X)) ≥ (g(X)). Then the following assertions hold.

	
  * (f(X)g(X))^⋆=f(X)^⋆∗ g(X)^⋆
	
  * (f(X)+g(X))^⋆=f(X)^⋆ + g(X) ∗ X^(f(X))-(g(X)).


	
A code is reversible complement if _^c ∈ C for any ∈ C. The next theorem characterizes reversible complement R-skew cylic code.
	

Let two polynomials g_1(X) and g_2(X) divide X^β-1 in _4[X]. Let C= ⟨ g(X)⟩ be R-skew cyclic with g(X)=v g_1(X) + (v+1) g_2(X). Then C is reversible complement if and only if g(X) is self-reciprocal and v (X^β-1)/(X-1) ∈ C.

	

Let g(X)=v g_1(X)+ (v+1) g_2(X) and C=⟨ g(X)⟩ be an R-skew cyclic code of length β. Suppose that C is reversible complement. Since ∈ C, we have 
(0,0,…,0) 
=(v,v,…,v) = v (X^β-1)/(X-1) ∈ C. Let 

    g_1(X)     = g_0 + g_1 X + … + g_t-1 X^t-1 + X^t
    
    g_2(X)     = h_0 + h_1 X + … + h_k-1 X^k-1 + X^k

where t ≤ k. Then

    g(X)    = v g_1(X) + (v+1) g_2(X) 
       = (v g_0 + (v+1) h_0) + (v g_1 + (v+1) h_1) X+ …
        + (v g_t-1 + (v+1) h_t-1) X^t-1 + (v+ (v+1) h_t) X^t
        + (v+1) h_t+1 X^t+1+ … + (v+1) h_k-1 X^k-1 + (v+1) X^k.

Since C is reversible complement, it contains   

    g_^c (X) 
        = v (1 + X + … + X^β-k-2) +  (v+1) 
    X^β- k -1  + (v+1) h_k-1 X^β-k
        + … + (v+1) h_t+1 X^β -t-2 + 
    (v+(v+1) h_t) X^β-t-1
        + (v g_t-1 + (v+1) h_t-1) X^β-t + … + 
    (v g_1  + (v+1) h_1) X^β-2
        + (v g_0 + (v+1) h_0) X^β-1.

Using Lemma <ref> we can write 

    g_^c (X)     =
    v (1+ X + … + X^β-k-2)+ (v+1) X^β -k-1 + (v+1) h_k-1 X^β -k
        + … + (v+1) h_t+1 X^β-t-2 + v 
    X^β-t-1 + (v+1) h_t X^β-t-1 + v X^β-t-1
        + v g_t-1 X^β-t + (v+1) h_t-1 X^β-t + v X^β-t + … + v g_1X^β-2
       + (v+1) h_1 X^β -2 + v x^β-2 + v g_0 X^β-1 + (v+1) h_0 X^β-1 + v X^β-1.

Because C is R-linear, g_^c (X) + v (X^β-1) / (X-1) ∈ C. This implies 

    g_^c (X) + v (X^β-1)/(X-1) 
    = 
       ((v+1) + v) X^β-k-1 + ((v+1) h_k-1 + v) X^β-k + … + 
        ((v+1) h_t+1 + v) X^β-t-2 + 
    (v + v) X^β-t-1+ ((v+1) h_t + v) 
    X^β-t-1 + 
        (v g_t-1 + v) X^β-t  + ((v+1) h_t-1 + v) 
    X^β-t + … + (v g_1 + v) X^β-2 + 
        ((v+1) h_1 + v) X^β -2 + (v g_0 + v) X^β-1 + ((v+1) h_0 + v) X^β-1.

By Equation (<ref>) we can write

    (v+1) X^β-k-1 + (v+1) h_k-1 X^β-k + … + (v+1) h_t+1 X^β-t-2 +
        v X^β-t-1+ (v+1) h_t X^β-t-1 + 
    X^β-t + (v+1) h_t-1X^β-t + … + 
        v g_1 X^β-2 + (v+1) h_1 X^β-2+ v g_0 X^β-1 + (v+1) h_0 
    X^β-1

as

    (v+1)  X^β-k-1 + (v+1) h_k-1 X^β-k + … + (v+1) h_t+1 X^β-t-2 + 
       (v + (v+1) h_t) X^β-t-1 + (v g_t-1 + (v+1) h_t-1) X^β-t+ … + 
       (v g_1 + (v+1) h_1) X^β-2 + 
    	(v g_0 + (v+1) h_0) X^β-1.

Multiplying on the right by X^k + 1 -β, we obtain

    (v+1) + (v+1) h_k-1θ(1) X + … + (v+1) h_t+1θ(1) X^k-t-1
    
    + (v+(v+1) h_t)θ(1) X^k-t + (v g_t-1 + (v+1) h_t-1)θ(1) X^k-t+1 + …
    
    + (v g_1 + (v+1) h_1)θ(1) X^k-1 + (v g_0 + (v+1) h_0)θ(1) X^k.

Hence, g^⋆(X) ∈ C. Since C = ⟨ g(X) ⟩, there exists q(X) ∈ R[X,θ] such that g^⋆(X)= q(X) ∗ g(X), which implies (g^⋆(X)) =  (g(X)) and q(X)=1. Thus, g^⋆(X) =g(X), as required.
		
Conversely, let C=⟨ g(X)⟩ be an R-skew cyclic code of length β generated by g(X) = v g_1(X) + (v+1) g_2(X) where g_1(X) and g_2(X) are two divisors of X^β-1 in _4[X]. Let c(X) = c_0 + c_1 X + … + c_k X^k∈ C, then there exist q(X) ∈ R[X,θ] such that c(X) = q(X) ∗ g(X). By Lemma <ref>, c^⋆(X) =q^⋆(X) ∗ g^⋆(X). Since C is self reciprocal, c^⋆(X) = q^⋆(X) ∗ g(X) ∈ C for any c(X)∈ C. Since C is skew cylic, c(X) ∗ X^β-k-1 = c_0 X^β-k-1 + c_1 X^β-k + … + c_k X^β-1∈ C. Hence, v (X^β-1) / (X-1) = 
v (1 + … + X^β-1) ∈ C. Since C is R-linear, 

    c(X) ∗ X^β-k-1 + v (X^β-1) / (X-1) =
    
    v + … + v X^β-k-2 + (c_0 + v) X^β-k-1 + … + (c_k + v) X^β-1.

By Equation (<ref>), 

    v + … + v X^β-k-2 + c_0 
    X^β-k-1 + … + c_K X^β-1
    =(c^∗(X))_^c ∈ C.

This concludes the proof.  

	
The theorem that we have just proved leads us from R-skew cyclic code to the definition and subsequent characterization of _4 R-skew cyclic code in the context of DNA coding.


An _4R-linear code C is DNA-skew cyclic if the followings hold.

	
  * C is an _4R-skew cyclic code, , C is an R-left submodule of 
	
    F[X]/(X^α-1) × R[X,θ ]/(X^β-1).

	
	
  * Any codeword =(_1,_2)∈ C and its reverse complement 
	
    _^c=((_1)_^c,(_2)_^c) ∈ C

	must be distinct.


	
The characterization of reverse complement codes over _4R can now be established. 

Let C=⟨ (f(X),),(,g(X) ⟩ be an _4R-skew cyclic code. Note that ℓ(X):= and C=C_1⊗ C_2 with C_1 an _4-cyclic code and C_2 an R-skew cyclic code. Then C is reversible complement if and only if C_1 and C_2 are reversible complement over _4 and R, respectively.

	

Let C be an _4R-skew cyclic code generated by (f(X),) and (,g(X)). Lemma <ref> shows how to find C=C_1⊗ C_2. Let =(_1,_2)∈ C = C_1⊗ C_2 with _1∈ C_1 and _2∈ C_2. Suppose that C_1 and C_2 are reversible complement over _4 and R, respectively. Then we have (_1)_^c∈ C_1 and (_2)_^c∈ C_2. Thus, ((_1)_^c, (_2)_^c)= _^c∈ C_1⊗ C_2=C.
		
Conversely, let _1∈ C_1 and _2∈ C_2. Then  =(_1,_2) ∈ C. If C is reversible complement, then   _^c=((_1)_^c,(_2)_^c) ∈ C = C_1⊗ C_2. This implies _1,^c∈ C_1 and _2,^c∈ C_2, as required. 




§ CONCLUSION


We have presented our studies on skew cyclic codes over the ring _4R. Their  algebraic structures as left submodules of a skew-polynomial ring are investigated, resulting in the identification of their generators. The fact that, under some simple conditions on their length, they are equivalent to cyclic or 2-quasi-cyclic codes over the same ring is established. Towards the end we show how the setup leads naturally to DNA codes and prove a condition on the associated generator polynomial of an _4R-skew cyclic code that guarantee the code to be reversible complement. We are now looking into whether the class of codes that we propose here contains those with better relative distance or size than known DNA codes.



§ ACKNOWLEDGMENTS

We would like to thank Jeremy Le Borgne for the MAGMA source code for factorization in _q[X,θ]. 

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Received xxxx 20xx; revised xxxx 20xx.


