Θ S cyclic codes over A k

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1 Θ cyclic codes over A k Irwansyah, Aleams Barra Institut Teknologi Bandung arxiv: v1 [cs.it] 15 Jul 2017 Bandung, Indonesia teven T. Dougherty University of cranton cranton, PA, UA Ahmad Muchlis, Intan Muchtadi-Alamsyah Institut Teknologi Bandung Bandung, Indonesia Patrick olé Telecom Paris Tech, Paris, France and University King Abdul Aziz, Jeddah, audi Arabia Djoko uprijanto Institut Teknologi Bandung Bandung, Indonesia Olfa Yemen Institut Préparatoire Aux Etudes D Ingénieurs El Manar, Tunis, Tunisia February 15, 2018 Abstract We study Θ cyclic codes over the family of rings A k. We characterize Θ cyclic codes in terms of their binary images. A family of Hermitian inner-products is defined and we prove that if a code is Θ cyclic then its Hermitian dual is also Θ cyclic. Finally, we give constructions of Θ cyclic codes. Corresponding author. djoko@math.itb.ac.id 1

2 Key Words: kew-cyclic codes; codes over rings. 1 Introduction Codes over rings are a widely studied object. At the heart of this subject is the construction of distance preserving maps from rings to the binary Hamming space. Initially, the four rings of order 4, F 4, Z 4, F 2 +uf 2,u 2 = 0 and F 2 +vf 2,v 2 = v were studied with respect to their distance preserving maps. ee [7], for a complete description of codes over these rings. The finite field F 4 and the ring Z 4 are part of the well known families of finite fields and integer modulo rings. Codes over these rings have been well studied. The ring F 2 +uf 2 has been generalized to the family of rings R k. ee [9], [10] and [11] for a description of codes over these rings. Codes over the ring F 2 +vf 2 have been studied in numerous papers, see [1], [6] and [7] for example. This family of rings has been generalized to the family of rings A k in [5]. Cyclic codes have long been one of the most interesting families of codes because of their rich algebraic structure. Namely, they can be viewed as ideals in a polynomial ring. This connection allows for a classification of cyclic codes by classifying ideals in a polynomial ring moded out by x n 1. In [3], skew cyclic codes were described as a generalization of cyclic codes. This notion was applied to codes over F 2 + vf 2 in [1]. In this work, they described generator polynomials of θ cyclic codes defined over this ring as well as the generator polynomials of their duals with respect to both the Euclidean and the Hermitian inner product. They also provided some examples of optimal θ cyclic self-dual codes with respect to the Euclidean and Hermitian inner product. Recently, in [12], Gao further generalized previous work in [1] and [3] to θ cyclic codes over the ring F p +vf p, with p prime. Among his results is that θ cyclic codes over the ring F p + vf p is equivalent to either cyclic codes or quasi-cyclic codes over the same ring. We note in [14], iap et.al. have proven the same result earlier for the case of θ cyclic codes over the field F q. In this work, we study skew cyclic codes over the family of rings A k and we study their images via a recursively defined distance preserving map into the binary Hamming space. 2 Definitions and Notations 2.1 Family of Rings In [5], the following family of rings, which are a generalization of the ring F 2 + vf 2, were defined. For the integer k 1, let A k = F 2 [v 1,v 2,...,v k ]/ v 2 i = v i,v i v j = v j v i. 2

3 The rings in this family of rings are finite commutative rings with cardinality 2 2k and characteristic 2. We shall describe a notation for any element in the ring A k. Let k be an integer with k 1. Let B {1,2,...,k} and let v B = i B v i. In particular, v = 1. It follows that each element of A k is of the form B P k α B v B where α B F 2, and P k is the power set of the set {1,2,...,k}. For A,B {1,2,...,k} we have that v A v B = v A B which gives that α B v B β C v C = α B β C v D. B P k C P k B C=D It is shown, in [5], that the only unit in the ring A k is 1. It is also shown that the ideal w 1,w 2,...,w k, where w i {v i,1+v i }, is a maximal ideal of cardinality 2 2k 1. Note that this gives 2 k maximal ideals. Hence, except for the case when k = 0, namely the finite field of order 2, the ring is not a local ring. The ring A k is a principal ideal ring. In particular, let I = α 1,α 2,...,α s be an ideal in A k, then I is a principal ideal generated by the element which is the sum of all non-empty products of the α i, that is I = α i. D P k A {1,2,...,s}, A Let be a subset of {1,2,...,k}. We shall define a set of automorphisms in the ring A k based on the set. Define the map Θ i by i A Θ i v i = v i +1 and Θ i v j = v j, j i. For all {1,2,...,k} the automorphism Θ is defined by: Θ = i Θ i. NotethatΘ isaninvolutionontheringa k. WeshallusethisinvolutiontodefineΘ cyclic codes. In the ambient space A n k we have two natural families of inner-products. First we have the standard Euclidean inner-product: [w,u] = w j u j. Now we define the Hermitian inner-product with respect to a subset T {1,2,...,k}. Define [w,u] HT = w j Θ T u j. Wehaveorthogonalscorresponding toeachinner-product. Wedefine C = {w [w,u] = 0, u C} and C H T = {w [w,u] HT = 0, u C}. Note that there are 2 k possible Hermitian duals for codes over A k. Moreover, we notice that if T = the Hermitian 3

4 orthogonal is in fact the Euclidean inner-product. We retain the Euclidean orthogonal notation because of its importance as a particular example of orthogonals. ince the ring A k is a Frobenius ring, in both cases we have the standard cardinality condition, namely C C = A n k and C CH T = A n k. Furthermore, we also have several properties of the ring A k as follows. ee [5], for foundational material on these rings. Proposition 2.1. If I is a maximal ideal in A k, then I = w 1,w 2,...,w k, where w i {v i,v i +1}, for 1 i k. Proof. ince A k is a principal ideal ring, we have I = ω, for some ω A k. Let ω = B P k α B v B for some α B F 2, v B = i B v i, and v = 1. If α = 0, then clearly ω v 1,v 2,...,v k. On the other hand, if α = 1, then let ω = 1+v B + B P k \{B, } α B v B where B P k such that α B = 1. We use mathematical induction on the cardinality of B. If B = {j 1 }, then ω v j1 +1,v j2,...,v jk. If B = {j 1,j 2 }, then consider v B +1 = v j1 +1v B +1+v j1 v B \{j 1 } +1 = v j1 +1v j1 v j2 +1+v j1 v j2 +1. As a consequence, ω v j1 +1,v j2 +1,v j3,...,v jk. Moreover, if B = {j 1,j 2,j 3 }, then By the previous equation we have v B +1 = v j1 +1v j1 v j2 v j3 +1+v j1 v j2 v j3 +1. v B +1 = v j1 +1v j1 v j2 v j3 +1+v j1 v j2 +1v j2 v j3 +1+v j2 v j3 +1, consequently, ω v j1 +1,v j2 +1,v j3 +1,v j4,...,v jk. Now, assume that the above equation is satisfied when B = m 1, then when B = m, we have v B +1 = v jm +1v B +1+v jm v B \{j m} +1. This implies that ω J = v j1 +1,...,v jm +1,v jm+1,...,v jk. ince J and I are maximal ideals, and I J, then we have that I = J. Lemma 2.2. The ring A k can be viewed as an F 2 -vector space with dimension 2 k whose basis consists of elements of the form i B w i, where B P k and w i {v i,v i +1}. 4

5 Proof. Every element a A k can be written as a = B P k α B v B, for some α B F 2, v B = i B v i and v = 1. Therefore, A k is a vector space over F 2 with a basis that consists of elements of the form v B = i B v i, where v = 1. There are k k j=0 j = 2 k basis elements. ince all v B are linearly independent over F 2, so is the element w B = i B w i, where w i {v i,1+v i }. As a direct consequence of the above proposition and [5, Theorem 2.3], we have the following theorem. Theorem 2.3. An ideal I in A k is maximal if and only if I = w 1,w 2,...,w k. The following proposition gives a characterization of automorphisms in A k. Proposition 2.4. Let θ be an endomorphism in A k. The map θ is an automorphism if and only if θv i = w j, where w j {v j,v j +1}, for every i {1,2,...,k}, θv i θv j when i j, and θa = a, for every a F 2. Proof. Let J = v 1,...,v k and J θ = θv 1,...,θv k. Consider the map λ : A k A k J J θ t+j θt+j θ. We can see that the map λ is a ring homomorphism. For any a,b A k /J where λa = λb, let a = a 1 + J and b = b 1 + J for some a 1,b 1 A k. We have that θa 1 b 1 J θ, so a 1 b 1 J. Consequently, a b = 0 + J, which means a = b, in other words, λ is a monomorphism. Moreover, for any a A k /J θ, let a = a 2 +J θ for some a 2 A k, then there exists a = θ 1 a 2 +J such that λa = a. Therefore, F 2 A k /J A k /J θ, which implies J θ is also a maximal ideal. By Theorem 2.3, J θ = w 1,w 2,...,w k. By [5, Theorem 2.6], J θ = w i = θv i B P k \{ } i B which means, B P k \{ } i B w i and B P k \{ } B P k \{ } i B i B θv i are associate. ince the only unit in A k is 1, then they must be equal. Therefore, θv i = w j. ince θ is an automorphism, θv i θv j whenever i j. uppose that θv i = w j, and θv i θv j whenever i j. By Lemma 2.2, we can see that θ is also an automorphism. We have the following immediate consequence. Corollary 2.5. An involution ψ of A k must be Θ or it must have ψv i = w j {v j,1+v j } and ψw j = v i where i,j {1,...,k}. Proof. By the previous proof we have that the only automorphisms send v i to either v j or 1+v j. Then to be an involution the application of the map twice must be the identity which gives the result. 5

6 2.2 Gray Map Gray maps were defined on all commutative rings of order 4, see [7] for a description of these four Gray maps. For the ring A 1 = F 2 +vf 2, we have the Gray map φ 1 : A 1 F 2 2 defined by φa+bv 1 = a,a+b. For A 1 this is realized as v 01 1+v 10. We extend this map inductively as follows. Every element in the ring A k can be written as α+βv k, where α,β A k 1. Then for k 2, define φ k : A k A 2 k 1 by φ k α+βv k = α,α+β. Then define Φ k : A k F 2k 2 by Φ 1 γ = φ 1 γ, Φ 2 γ = φ 1 φ 2 γ and Φ k γ = φ 1 φ 2...φ k 2 φ k 1 φ k γ... It follows immediately that Φ k 1 = 1, the all one vector. We note that the Gray map is a bijection and is a linear map. Example 1. For k = 2, we have that Φ 2 : A 2 F 4 2 and Φ 2 a+bv 1 +cv 2 +dv 1 v 2 = a,a+b,a+c,a+b+c+d. Let T be the matrix that performs the cyclic shift onavector. That is Tv 1,v 2,...,v n = v n,v 1,...,v n 1. Let σ i,k be the permutation on {1,2,...,2 k } defined by σ i,k {p2 i +1,...,p+12 i } = T 2i 1 p2 i +1,...,p+12 i, for all 0 p 2 k i 1. Let Σ i,k be the permutation on elements of F 2k 2 induced by σ i,k. That is, for x = x 1,x 2,...,x 2 k F 2k 2, Σ i,k x = x σi,k 1,x σi,k 2,...,x σi,k 2 k. 1 In other word, Σ i,k is a permutation induced by σ i,k. To clarify the permutation above, let us consider an example below. Example 2. Let k = 2 and i = 1. We want to find Σ 1,2 x, for all x = x 1,x 2,x 3,x 4 F ince i = 1, we have 0 p 1. Hence, for p = 0 we have σ 1,2 {1,2} = T1,2 = 2,1 6

7 and for p = 1 we have σ 1,2 {3,4} = T3,4 = 4,3. Therefore, For i = 2, we have p = 0. Then, Σ 1,2 x = x σ1,2 1,x σ1,2 2,x σ1,2 3,x σ1,2 4 = x 2,x 1,x 4,x 3. σ 2,2 {1,2,3,4} = T 2 1,2,3,4 = 3,4,1,2. Therefore, we have Σ 2,2 x = x σ2,2 1,x σ2,2 2,x σ2,2 3,x σ2,2 4 = x 3,x 4,x 1,x 2. We can also define another map which will be used later for constructing generators for Θ cyclic codes and optimal codes. Let p,k N, where p < k. Let Ω p = {p + 1,...,k} and s = 2 k p. We have that P k Ω p = s. We can define a Gray map as follows: Ψ k,p : A k A s p. Denote the coordinates of A s p by the lexicographic ordering of the subsets of Ω p and denote them by B 1,...,B s. Note that B 1 = and B s = Ω p. An element of A k can be written as B Ω p α B w B, where α B A p and w B = i B v i. Then Ψ k,p α B w B = α D, α D,..., α D. B Ω p D B 1 D B 2 D B s For p = 0, Ψ k,0 is the same map as Ψ k [5]. In that paper, it is shown that Ψ k and Φ k are conjugate, i.e. their images are permutation equivalent. Notice that the representation of an element in A k can be changed by replacing any v i with 1 + v i. In that way, we can let u i be either v i or v i + 1 for each i. Then we have an alternative definition of Ψ k,p as Ψ k,p α B y B = α D, α D,..., α D B Ω p D B 1 D B 2 D B s where y B = i B u i. Any result for Ψ k,p can be replaced for Ψ k,p. Lemma 2.6. Let k 1 and 1 i k. For x A k we have Σ i,k Φ k x = Φ k Θ i x. 7

8 Proof. Let 1 i k. For k = 1, we have x = a + bv and Θ 1 x = a + b + bv. Thus Φ 1 Θ 1 x = a+b,a = Σ 1,1 Φ 1 x. Assume the result is true for values less than or equal to k 1 and let i < k with x A k. We have x = a+bv k with a,b A k 1 and Θ i x = Θ i a+θ i bv k. Then It follows that φ k x = a,a+b. Φ k x = Φ k 1 a,φ k 1 a+b and Φ k Θ i x = Φ k 1 Θ i a,φ k 1 Θ i a+θ i b. We note that and Φ k 1 a = x 1,x 2,...,x 2 k 1, Φ k 1 a+b = x 1,x 2,...,x 2 k 1 Φ k x = Φ k 1 a,φ k 1 a+b = y 1,y 2,...,y 2 k 1,y 2 k 1 +1,...,y 2 k withx 1,x 2,...,x 2 k 1 = y 1,y 2,...,y 2 k 1andx 1,x 2,...,x 2 k 1 = y 2 k 1 +1,y 2 k 1 +2,...,y 2 k. We have that Σ i,k Φ k x = Σ i,k y 1,...,y 2 i, y }{{} +1,...,y 2.2i,...,y } 2i {{} 2k i 12i+1,...,y 2k }{{} 2 i 2 i 2 i = y 2 i 1 +1,y 2 i 1 +2,...,y 2 i,y 1,...,y 2 i 1, }{{} 2 i y 2 i+1 2 i 1 +1,y 2 i+1 2 i 1 +2,...,y 2 i +1,...,y 2 i+1 2 }{{ i 1, } 2 i...,y 2 i ,y k i+1 2 i ,...,y k i+1 2 k,...,y 2 i 1 2 k i+1 1 }{{} 2 i = Σ i,k 1 Φ k 1 a,σ i,k 1 Φ k 1 a+b. The hypothesis of recurrence implies that Σ i,k 1 Φ k x = Φ k 1 Θ i a,φ k 1 Θ i b = Φ k Θ i x. Now consider the case when i = k. For x A k, x = a+bv k with a,b A k 1 we have that Θ k x = a+b+bv k. Then φ k x = a,a+b, φ k Θ k x = a+b,a and consequently Φ k x = Φ k 1 a,φ k 1 a+b and Φ k Θ k x = Φ k 1 a+b,φ k 1 a+b. 8

9 We note that Φ k 1 a = x 1,...,x 2 k 1, Φ k 1 a+b = x 1,...,x 2 k 1 and Φ k x = Φ k 1 a,φ k 1 a+b = y 1,y 2,...,y 2 k 1,y 2 k 1 +1,...,y 2 k with x 1,...,x 2 k 1 = y 1,...,y 2 k 1 and x 1,...,x 2 k 1 = y 2 k 1 +1,...,y 2 k. On the other hand, when i = k we have that Then we have that Σ k,k = T 2k 1. Σ k,k Φ k x = Σ i,k y 1,...,y 2 k 1,y 2 k 1 +1,...,y 2 k = T 2k 1 y 1,...,y 2 k 1,y 2 k 1 +1,...,y 2 k = y 2 k 1 +1,...,y 2 k,y 1,...,y 2 k 1 = Φ k 1 a+b,φ k 1 a = Φ k Θ k x. Example 3. Consider Then also, Φ 2 v 1 v 2 = Φ 1 0,Φ 1 v 1 = 0,0,0,1. Φ 2 Θ 1 v 1 v 2 = Φ 2 v 1 +1v 2 = Φ 1 0,Φ 1 1+v 1 = 0,0,1,0 = Σ 1,2 0,0,0,1 = Σ 1,2 Φ 2 v 1 v 2, Φ 2 Θ 2 v 1 v 2 = Φ 2 v 1 +v 1 v 2 = Φ 1 v 1,Φ 1 0 = 0,1,0,0 = Σ 2,2 0,0,0,1 = Σ 2,2 Φ 2 v 1 v 2. We can extend the definition of Σ to subsets of {1,2,...,k}. 9

10 Definition 1. For all A {1,2,...,k} we define the permutation Σ A,k by Σ A,k = i AΣ i,k. It is clear that for all x A k we have Σ A,k Φ k x = Φ k Θ A x. 2 3 Θ cyclic codes over A k We can now define skew cyclic codes using this family of rings and family of automorphisms. Definition 2. A subset C of A n k is called a skew cyclic code of length n if and only if then C is a submodule of A n k, and C is invariant under the Θ shift T Θ. The second condition can be seen as follows. If c = c 0,c 1,...,c n 1 C T Θ c = Θ c n 1,Θ c 0,...,Θ c n 2 C. We can now describe the ambient algebraic space we shall use to describe skew cyclic codes. Definition 3. Let {1,2,...,k}. Define the skew polynomial ring A k [x,θ ] = {a 0 + a 1 x+a 2 x 2 + +a n x n a i A k } as the set of polynomials over A k where the addition is the usual polynomial addition and the multiplication is defined by the basic rule xa = Θ ax and extended to all elements of A k [x,θ ] by associativity and distributivity. It is clear that the set A k n = A k [x,θ ]/ x n 1 is a left A k [x,θ ]-module. The next theorem follows immediately from the definitions. Theorem 3.1. A code C in A n k is a Θ cyclic code if and only if C is a left A k [x,θ ]- submodule of the left A k [x,θ ]-module A k n. The following theorem characterizes skew-polynomial rings over A k. Theorem 3.2. Let A k [x,θ 1 ] and A k [x,θ 2 ] be two skew polynomial rings over A k. Then, A k [x,θ 1 ] = A k [x,θ 2 ] if and only if 1 = 2. 10

11 Proof. Let λ be an automorphism on A k. By Theorem 3 in [13] and properties in [5], we have λ : x b 0 +b 1 x+ +b n x n can be extended to an isomorphism between A k [x,θ 1 ] and A k [x,θ 2 ] if and only if all the following conditions hold: λθ 1 ab i = b i Θ i 2 λa for every a A k and for all i = 0,1,...,n, b 1 = 1, b i = 0 for all i = 2,3,...,n. This implies that x x+b 0. It follows that we only need to check thefollowing conditions: and λθ 1 ab 0 = b 0 λa λθ 1 a = Θ 2 λa. For the first condition, if we choose a = v i where i 1, then we have λ1+v i v i b 0 = 0, which gives b 0 = 0. Moreover, the second condition gives Θ 2 = λθ 1 λ 1, and consequently, Θ 1 and Θ 2 are conjugate to each other. Which means, they have to has the same cycle structures as an automorphism. This gives 1 = 2, and the result follows. As an immediate consequence, we have the following result. Corollary 3.3. Let {1,2,...,k}. There are k skew-polynomialrings Ak [x,θ ] which are isomorphic to a given A k [x,θ ]. Proof. It is immediate since the number of subsets of {1,...,k} with cardinality is k. 4 Characterizations of Θ cyclic codes 4.1 First characterization Let {1,2,...,k} and let Σ = τ T 2k be the permutation on elements of F n2k 2 where T is the cyclic shift modulo n2 k and τ is the permutation on elements of F n2k 2 defined for all by x = x 1 1,...,x1 2 k,x2 1,...,x2 2 k,...,xn 1,...,xn 2 k F n2k, 2 τ x = τ x 1 1,...,x1 2 k,x 2 1,...,x2 2 k,...,x n 1,...,xn 2 k = Σ,k x 1,Σ,k x 2,...,Σ,k x n where x j = x j 1,...,x j. ince T 2k and τ 2 k commute, Σ can be written as T 2k τ as well. Let σ denote permutation on {1,2,...,n2 k }, the indices of elements in F n2k 2, that induce the permutation Σ above. 11

12 Lemma 4.1. Let C be a code in A n k. The code Φ kc is fixed by the permutation Σ if and only if C is a Θ cyclic code. Proof. Let C be a Θ cyclic code and let y = y 1,y 2,...,y n2 k Φ k C. That is, there exists x = x 1,x 2,...,x n C such that y = Φ k x 1,Φ k x 2,...,Φ k x n and Φ k x i F 2k 2. We have that Σ y = τ T 2k Φ k x 1,Φ k x 2,...,Φ k x n = τ Φ k x n,φ k x 1,...,Φ k x n 1 = Σ,k Φ k x n,σ,k Φ k x 1,...,Σ,k Φ k x n 1 = Φ k Θ x n,φ k Θ x 1,...,Φ k Θ x n 1 = Φ k Θ x Φ k C, since C is a Θ cyclic code. LetC beacodeinf n2k 2 thatisfixedbythepermutationσ andletx = x 1,x 2,...,x n Φ 1 k C. We have that there exists y = x 1 1,...,x 2k 1,x 1 2,...,x 2k 1,...,x 1 n,...,x 2k n C, such that y = Φ k x 1,x 2,...,x n = Φ k x 1,Φ k x 2,...,Φ k x n. ince C is fixed by the permutation Σ and Φ k x i F 2k 2, we have that for Σ y C : Σ y = τ T 2k Φ k x 1,Φ k x 2,...,Φ k x n = τ Φ k x n,φ k x 1,...,Φ k x n 1 = Σ,k Φ k x n,σ,k Φ k x 1,...,Σ,k Φ k x n 1 = Φ k Θ x n,φ k Θ x 1,...,Φ k Θ x n 1 = Φ k Θ x. It follows that Φ k Θ x C and thus Φ 1 k C is a Θ cyclic code. Next, we consider the order of the permutation σ which we will need in Theorem 4.3. Lemma 4.2. Let ordσ denote the order of σ. Then { n n 0 mod 2 ordσ = 2n n 1 mod 2. 3 Proof. It is clear that for n odd, the permutation σ is a product of 2 k 1 cycles of length 2n and we have σ = 1,σ 1,σ1,...,σ 2 2n 1 1 2,σ 2,σ 2 2,...,σ2n k 1,σ 2 k 1,σ 2 2 k 1,...,σ 2n 1 2 k 1. 12

13 Furthermore, if n is even, we have σ is a product of 2 k cycles of length n and σ = 1,σ 1,σ 2 1,...,σn 1 1 2,σ 2,σ 2 2,...,σn k,σ 2 k,σ 2 2 k,...,σ n 1 2 k. Theorem 4.3. Let C be a code in A n k. 1. If n is odd then C is a skew-cyclic code if and only if Φ k C is equivalent to an additive 2 k 1 quasi-cyclic code C in F n2k If n is even then C is a skew-cyclic code if and only if Φ k C is equivalent to an additive 2 k quasi-cyclic code C in F n2k 2. Proof. Let C be a skew-cyclic code in A n k. Consider the case when n is odd, where the equation in Lemma 4.2 holds. Define the permutation σ 1 by k 1 2 k k... 2n 12 k n2 k σ 1 = σ 2n σ 2n 1 2 k 1 σ 2n σ 2n 2 2 k 1... σ σ0 2k 4 It is clear that for all 1 j n2 k such that j = a2 k 1 + b with 0 a 2n 1 and 1 b 2 k 1 we have σ 1 j = σ 2n 1 a b. 5 The permutation σ 1 induces the permutation Σ 1 acting on the elements of F n2k 2. For v = v 1,v 2,...,v n2 k F n2k 2, Σ 1 v = v σ1 1,v σ1 2,...,v σ1 n2 k. 6 To show that Σ 1 Φ k C is a 2 k 1 quasi-cyclic code, we must prove that for all codewords v Φ k C, we have that T 2k 1 Σ 1 v Σ 1 Φ k C where T is the vector cyclic shift. ince Σ Φ k C = Φ k C by Lemma 4.2, we only need to show that T 2k 1 Σ 1 v = Σ 1 Σ v. 7 We start with the right hand side of the equation. By definition, Σ v = v σ 1,v σ 2,...,v σ 2n := v 1,v 2,...,v n2 k = v. 8 13

14 Applying σ 1 and by Equation 4, we have Σ 1 Σ v = v σ 1 1,v σ 1 2,...,v σ 1 2 k 1,v σ 1 2 k 1 +1,...,v σ 1 2 k 1,v σ 1 2n 12 k 1 +1,...,v σ 1 n2 k = 1,v σ 2n 1 2,...,v σ 2n 1 2 k 1,v σ 2n 2 1,...,v σ 01,...,v σ 02k 1. v σ 2n 1 Now, Equation 8 allows us to write Σ 1 Σ v = v σ 2n 1,v σ 2n 2,...,v σ 2n 2,v k 1 σ 2n 1 1,...,v σ 1,...,v σ 2 k 1 = v 1,v 2,...,v 2 k 1,v σ 2n 1 1,...,v σ 2n 1 2 k 1,...,v σ 1,...,v σ 2 k 1 by Lemma 4.2. ince Σ 1 v = v σ1 1,v σ1 2,...,v σ1 2,v k 1 σ1 2 +1,...,v k 1 σ1 2,...,v k σ1 2n 12 +1,...,v k 1 σ1 n2 k = v σ 2n 1 1,v σ 2n 1 2,...,v σ 2n 1 2 k 1,v σ 2n 2 1,...,v σ 2n 2 2 k 1,...,v σ 01,...,v σ 0 2 k 1 = v σ 2n 1 1,v σ 2n 1 2,...,v σ 2n 1 2 k 1,v σ 2n 2 1,...,v σ 2n 2 2 k 1,...,v 1,...,v 2 k 1 and Σ Φ k C = Φ k C, we get T 2k 1 Σ 1 v = Σ 1 Σ v Σ 1 Φ k C. Thus, Σ 1 Φ k C is a 2 k 1 quasi-cyclic code. Now let n be an even integer. Define the permutation σ 2 by k 2 k k+1... n 12 k n2 k σ 2 = σ n 1 1 σ n σ n 1 2 k σ n σ n 2 2 k... σ σ0 2k 9 Let us denote by Σ 2 the permutation induced by σ 2 acting on the elements of F n2k 2. For v = v 1,v 2,...,v 2n F n2k 2, Σ 2 v = v σ2 1,v σ2 2,...,v σ2 n2 k. 10. It is clear that for all 1 j n2 k, j = a2 k +b with 0 a n 1 and 1 b 2 k we have that σ 2 j = σ n 1 a b. 11 To show that Σ 2 Φ k C is a 2 k quasi-cyclic code we must prove that for all codewords v Φ k C, T 2k Σ 2 v Σ 2 Φ k C 14

15 where T is the vector cyclic shift. ince Σ Φ k C = Φ k C by Lemma 4.1, we only need to show that T 2k Σ 2 v = Σ 2 Σ v. 12 We have that Σ v = v σ 1,v σ 2,...,v σ 2n := v 1,v 2,...,v n2 k = v. 13 Applying σ 2 and by Equation 10, we have Σ 2 Σ v = v σ 2 1,v σ 2 2,...,v σ 2 2 k,v σ 2 2 k +1,...,v σ 2 2 k,v σ 2 n 12 k +1,...,v σ 2 n2 k = v σ n 1 1,v σ n 1 2,...,v σ n 1 2 k,v σ n 2 1,...,v σ 01,...,v σ 02k Equation13 allows us to write Σ 2 Σ v = v σ n 1,v σ n 2,...,v σ n 2,v k σ n 1 1,...,v σ 1,...,v σ 2 k = v 1,v 2,...,v 2 k,v σ n 1 1,...,v σ n 1 2 k,...,v σ 1,...,v σ 2 k by Lemma 4.2. ince Σ 2 v = v σ2 1,v σ2 2,...,v σ2 2,v k σ2 2 +1,...,v k σ ,v k σ2 n 12 +1,...,v k σ2 n2 k = v σ n 1 1,v σ n 1 2,...,v σ n 1 2 k,v σ n 2 1,...,v σ n 2 2 k,...,v σ 01,...,v σ 0 2 k = v σ n 1 1,v σ n 1 2,...,v σ n 1 2 k,v σ n 2 1,...,v σ n 2 2 k,...,v 1,...,v 2 k and Σ Φ k C = Φ k C, we get T 2k Σ 2 v = Σ 2 Σ v Σ 2 Φ k C. Thus, Σ 2 Φ k C is a 2 k quasi-cyclic code. This completes the proof. We provide below two examples of the first characterization of Θ cyclic codes. Example 4. Let C 1 = {0000,1010,1000,0010}. We can see that C is a binary 2 quasicyclic code of length 4. Now, consider Φ = 0,0, Φ = φ ,φ = 1+v,1+v, Φ = φ ,φ = 1+v,0, Φ = φ ,φ = 0,1+v. 15

16 Then we have Φ 1 1 C 1 = {0,0,1+v,1+v,1+v,0,0,1+v} which is a Θ cyclic code over A 1. Example 5. Let C 2 = 1,1,1 be a Θ {1,2} cyclic code of length 3 over A 2. Consider, Φ = φ 1 1,φ 1 1,φ 1 1,φ 1 1,φ 1 1,φ 1 1 = , Φ 2 v 1 v 1 v 1 = φ 1 v 1,φ 1 v 1,φ 1 v 1,φ 1 v 1,φ 1 v 1,φ 1 v 1 = , Φ 2 v 2,v 2,v 2 = φ 1 0,φ 1 1,φ 1 0,φ 1 1,φ 1 0,φ 1 1 = , Φ 2 v 1 v 2,v 1 v 2,v 1 v 2 = φ 1 0,φ 1 v 1,φ 1 0,φ 1 v 1,φ 1 0,φ 1 v 1 = We can see that, Φ 2 C 2 = , , , which is a binary 2 quasi-cyclic code. 4.2 econd characterization We also have a characterization of Θ cyclic codes using the map Ψ k,p as follows. Theorem 4.4. An A k -linearcodec isθ cyclic of lengthnif andonlyif C = Ψ 1 k,p C 1,...,C s, where C 1,...,C s are quasi-cyclic codes of index 2 which satisfy for some and permutation µ. T Θ C i C µi 14 Proof. Let = {i 1,...,i t }, then take p = i j 1 for some i j. We will have codes C 1,...,C s over A p such that C = Ψ 1 k,p C 1,...,C s. It is clear that C 1,...,C s are quasicyclic codes of index 2 since Θ is an involution. Now, for any i, let c i C i, and write c i = α 1 D,..., α n D. D B i D B i Consider, c i = Ψ k,p0,...,0,c i,0,...,0 = α 1 D D B i B Ω p,b B i w B,..., α n D D B i B Ω p,b B i w B and T Θ c i = Θ α n D Θ w B,..., Θ α n 1 D Θ w B D B i B Ω p,b B i D B i B Ω p,b B i C 16

17 where = {i 1,...,i j 1 }. When we consider Ψ k,p Θ w B, we can think of it is a permutation of Ψ k,p w B as in Equation 2, which gives permutation µ. Therefore, we have Ψ k,p T Θ c i = 0,...,0,T Θ c i,0,...,0 where T Θ c i C µi, as we hope. Using the above setting for p,, and, for any c C, let c = B,..., B. By Equation 2, we have Ψ k,p Θ c = which is in C 1,...,C s by assumption. B Ω p α 1 B Ω p α n D B µ1 Θ α 1 D,..., D B µs Θ α 1 D. D B µ1 Θ α n D,..., D B µs Θ α n D Two examples for the second characterization of Θ cyclic codes are given below. Example 6. Let C 3 = v 2,0,0,1+v 2,v 2,1+v 2. We can check that C 3 is a Θ {1,2} cyclic code of length 2 over A 2. Consider, Ψ 2,1 v 2,0 = 0,1,0,0, Ψ 2,1 0,1+v 2 = 0,0,1,0, Ψ 2,1 v 2,1+v 2 = 0,1,1,0, Ψ 2,1 v 1 v 2,0 = 0,v 1,0,0, Ψ 2,1 0,v 1 +v 1 v 2 = 0,0,v 1,0, Ψ 2,1 v 1 v 2,v 1 +v 1 v 2 = 0,v 1,v 1,0. We have that C 3 = Ψ 1 2,1C 4,C 5, where C 4 = 0,1 and C 5 = 1,0 whichare 2 quasicyclic codes over A 1. Note that, µ1 = 2 and µ2 = 1. Finally, we can see that T Θ{1} C 1 = C 2 = C µ1 and T Θ{1} C 2 = C 1 = C µ2. Example 7. Let C 6 = C 7 = {00,11} are binary cyclic codes. Consider, Ψ 1 1,0 00,11 = Ψ 1 1,0 01,Ψ 1 1,0 01 = v,v, Ψ 1 1,0 11,00 = Ψ 1 1,0 10,Ψ 1 1,0 10 = 1+v,1+v, Ψ 1 1,0 11,11 = Ψ 1 1,0 11,Ψ 1 1,0 11 = 1,1. o, we have Ψ 1 1,0C 6,C 7 = {0,0,v,v,1 + v,1 + v,1,1} which is a Θ {1} cyclic code over A 1. 17,

18 5 Construction of Θ cyclic codes over A k In this section, we illustrate some constructions of skew cyclic codes over A k. Recall that the ring A k is a principal ideal ring. As such it is isomorphic to a direct product of chain rings. In particular, A k is isomorphic as a ring to F 2k 2 via the Chinese Remainder Theorem, see [5] for a complete description of this. Let CRT : F 2k 2 A k be this canonical map. Let CRTC 1,...,C 2 k be the code over A k formed by taking the map from C 1 C 2 C 2 k where each C i is a binary code. Define the following map Γ : F n2k 2 F n 2 Fn 2 Fn 2 by Γx 1 1,...,x 2k 1,x 1 2,...,x 2k 1,x 1 3,...,x 1 n,...,x 2k n = x 1 1,...,x 1 n,x 2 1,...,x 2 n,...,x 2k 1,...,x 2k n. For all codes C over A k with C = CRTC 1,...,C 2 k we have that Γ Φ k C = C 1,C 2,...,C 2 k. Proposition 5.1. Let n be an even integer and let C 1,C 2,...,C 2 k be binary cyclic codes in F n 2. Then for all A {1,2,...,k} there exists a Θ A cyclic code C in A n k. Proof. It is clear that C = Γ 1 C 1,C 2,...,C 2 k is a 2 k quasi-cyclic code over F n2k 2. It follows from Theorem 4.3 that Φ 1 k σ 1 2 C is a Θ A cyclic code C in A n k. with We define the map Γ 1 : F n2k 1 2 F 2n 2 F 2n 2 F 2n 2, 15 Γ 1 x 1 1,...,x2k 1 1,x 1 2,...,x2k 1 2,...,x 1 2n,...,x2k 1 2n = x 1 1,...,x1 2n,x2 1,...,x2 2n,...,x2k 1 1,...,x 2k 1 2n. Proposition 5.2. Let n be an odd integer and let C 1,C 2,...,C 2 k 1 be binary cyclic codes in F 2n 2. Then for all {1,2,...,k} there exists a Θ cyclic code C in A n k. Proof. It is clear that C = Γ 1 1 C 1,...,C 2 k 1 is a 2 k 1 quasi-cyclic code over F n2k 2. It follows from Theorem 4.3 that Φ 1 k σ 1 1 C is a Θ cyclic code C in A n k. We know describe an algorithm for constructing Θ cyclic codes. 1. Construction of Θ A cyclic codes in A k of even length a We consider C 1,...,C 2 k binary cyclic codes in F n 2 b We apply Γ 1 and we obtain C a 2 k quasi-cyclic code in F n2k 2. c We apply Φ 1 k σ 1 2 to C. We obtain a Θ cyclic code C in A k. 18

19 2. Construction of Θ cyclic codes in A k of odd length a We consider C 1,...,C 2 k 1 binary cyclic codes in F 2n 2 b We apply Γ 1 1 and we obtain C a 2 k 1 quasi-cyclic code in F n2k 2. c We apply Φ 1 k σ 1 1 to C. We obtain a Θ cyclic code C in A n k. 3. Construction of Θ cyclic codes over A k from codes over A p, where p < k a Given C 1,...,C s quasi-cyclic codes of index 2 in A p which satisfy Equation 14 in Theorem 4.4, for some. b Appling Ψ k,p to C 1,...,C s, we obtain a Θ cyclic code over A k. In terms of skew-polynomial rings, the third construction of a Θ cyclic code above will be as follows. Proposition 5.3. Let C = Ψ 1 k,p C 1,...,C s be Θ cyclic codes over A k, where C 1,...,C s are codes over A p, for some p < k. If C i = g 1i x,...,g mi x, for all i = 1,...,s, then C = g 11 x,...,g m1 x,...,g 1s x,...,g ms x. Proof. For any cx C, there exist c i x C i, where 1 i s, such that cx = Ψ 1 k,p c 1x,...,c s x. Now, let m j c j x = α kj xg jk for some α kj x A p [x,θ ], for all j = 1,...,s. Then we have m1 cx = w B1 α k1 xg 1k as we hope. k=1 k=1 m i + +w Bi α ki xg ik k=1 ms + +w Bs α ks xg sk k=1 mj B j B i k=1 mj s j=1 α kj xg jk α kj xg jk k=1 19

20 6 elf-dual codes In this section we will give a characterization for Hermitian self-dual codes over A k. FirstwewillshowthattheorthogonalofaΘ cyclic codeisagainaθ cyclic code. We note that the Euclidean inner-product is simply the Hermitian inner-product with R =. Hence any proof for the Hermitian inner-product applies as well to the Euclidean innerproduct. Theorem 6.1. If C is a Θ cyclic code then C H R is a Θ cyclic code for all R {1,2,...,k}. Proof. Let,R {1,2,...,k}. Let u C and w C H R. We know that since u is an element of C that T i Θ u C for all i. Hence we have that [T i Θ u,w] HR = 0. Then consider Θ Θ R [T i Θ u,w] HR = 0. Θ Θ R [TΘ i u,w] HR = Θ Θ R T i Θ u j Θ R w j = Θ Θ R Θ u j T i Θ R w j i = Θ R u j T i Θ w j i = 0. This gives that T i Θ w C H R and so C H R is Θ cyclic. Next, we have the following lemma. Lemma 6.2. A code C over A k is self-dual if and only if C = CRTC 1,...,C 2 k and C i = C i for all i. Proof. Follows from Theorem 6.4 in [8]. Then we also have the following characterization. Theorem 6.3. A code C over A k satisfies C = C H T if and only if C = CRTC 1,...,C 2 k and C i = C µi, for some permutation µ, for all i. Proof. Recall that C H T = Θ T C. By Equation 2, we can say that the map Θ T induces a permutation µ on the coordinates of Φ k A k. o, if C = CRTC 1,...,C 2 k then and the result follows. Θ T C = CRTC µ1,...,c µ2 k, As an easy consequence, we have the following corollary. 20

21 Corollary 6.4. A Θ cyclic code C is a Hermitian self-dual code with respect to Θ T if and only if C = CRTC 1,...,C 2 k, C i = C µi for all i, where µ is a permutation induced by Θ T, and Φ k C is fixed by the permutation Σ. Proof. Follows from Lemma 4.1 and Theorem 6.3. Here is an example of a self-dual code. Example 8. Let C 8 = {0,0,v,v,1 + v,1 + v,1,1} is a Θ {1} cyclic code over A 1. Furthermore, if T = {1}, then C H T 8 = C 8. Using the calculation in Example 7, we have that C 8 = Ψ 1 1,0C,C, where C = C = {00,11} which are two binary cyclic self-dual codes. 7 Optimal codes Inthissection, wewill giveawaytoconstruct optimalθ cycliccodesusingproposition5.3 and some examples of optimal codes obtained using this technique. First, we need the following lemma. Lemma 7.1. Let C be a linear code over A k. Then, if C = Ψ 1 k,p C 1,...,C s, for some codes C 1,...,C s over A p, for some p, then d H C = min 1 i s d H C i. Proof. Follows from Lemma 6.2 in [8]. ThismeansthatwecanusetheoptimalbinarycodesoroptimalΘ cyclic codesover A p to construct optimal Θ cyclic codes over A k for all k with respect to the Hamming weight. There are tables for binary optimal Euclidean self-dual codes in[4] and for optimal Hermitian self-dual codes and Hermitian Type IV self-dual codes in [2]. Therefore, in particular, we only need the optimal binary quasi-cyclic codes to construct self-dual optimal Θ cyclic codes over A k using the result in Proposition 5.3 or the algorithm described in ection 5. For example, we can use Θ cyclic codes over A 1 in [1] to produce the optimal Θ cyclic codes over A k for k 2, as in Table 1. n d Generator polynomial A k T 4 2 x 2 +1 A 2 {1,2} {1} 6 2 x 3 +1 A 7 {1,3,5,7} {1} 8 4 x 4 +v 1 +1x 3 +x 2 +v 1 x+1 A 2 {1,2} 10 2 x 5 +1 A 3 {1,2} 12 4 x 6 +v 1 +1x 5 +x 4 +x 3 +x 2 +v 1 x+1 A 3 {1,3} 14 4 x 7 +x 6 +x 5 +x 4 +x+1 A 4 {1,2,3} {1} 16 4 x 8 +v 1 +1x 5 +x 4 +v 1 x 3 +1 A 4 {1,2,3} {1} 18 4 x 9 +x 7 +v 1 x 6 +1+v 1 x 5 +1+v 1 x 4 +v 1 x 3 +x 2 +1 A 2 {1,2} {1} 21

22 20 4 x 1 0+v 1 +1x 7 +x 6 +x 5 +x 4 +v 1 x 3 +1 A 6 {1,2,6} 22 6 x 11 +x 10 +v 1 x 9 +v 1 x 8 +v 1 x 7 +v 1 +1x 6 +v 1 +1x 5 +v 1 x 4 +v 1 x 3 +v 1 x 2 +x+1 A 3 {1,2} {1} 24 8 x 12 +x 11 +v 1 x 10 +x 9 +v 1 +1x 7 +v 1 x 5 +x 3 +v 1 +1x 2 +x+1 A 5 {1,3,4,5} 26 6 x 13 +x 11 +v 1 x v 1 x 9 +v 1 x 8 +v 1 x 7 +1+v 1 x 6 +1+v 1 x 5 +v 1 x 4 +1+v 1 x 3 +x 2 +1 A 2 {1,2} {1} Table 1: Table of examples of optimal self-dual Θ cyclic codes Acknowledgement A part of the work of D.. was done when he visited Research Center for Pure and Applied Mathematics RCPAM, Graduate chool of Information ciences, Tohoku University Japan, on June 2015 under the financial support from Riset Desentralisasi ITB-Dikti 2015 Number 311c/I1.C01/PL/2015. A.B., I., and I.M.-A. were supported in part by Riset dan Inovasi KK ITB tahun References [1] T. Abualrub, N.Aydin, P. eneviratne, OnΘ cyclic codesover F 2 +vf 2, Austrl. Journal of Comb. 54, 2012, [2] K. Betsumiya, and M. Harada, Optimal elf-dual Codes over F 2 F 2 with Respect to The Hamming Weight, IEEE Trans. Inform. Theory 50 2, 2004, [3] D. Boucher, W. Geiselmann and F. Ulmer, kew-cyclic codes, Appl. Algebra in Eng., Commun. and Comp. 18, 2007, [4] J.H. Conway, and N.J.A loane, A New Upper Bound on the Minimal Distance of elf- Dual Codes, IEEE Trans. Inform. Theory 36 6, 1990, [5] Y. Cengellenmis, A. Dertli, and.t. Dougherty, Codes over an infinite family of rings with a Gray map, Des., Codes and Cryptog. 72 3, 2014, [6] A. Dertli, Y. Cengellenmis, MacDonald codes over the ring F 2 +vf 2. Int. J. Algebra 5, 2011,

23 [7].T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, and P. olé. Type IV elf-dual Codes over Rings, IEEE Trans. Inform. Theory 45 7, November 1999, [8].T. Dougherty, J.L. Kim, and H. Kulosman, MD Codes over Finite Principal Ideal Rings, Des., Codes, and Cryptog. 50 1, 2009, [9].T. Dougherty, B. Yildiz, and. Karadeniz, Codes over R k, Graymapsandtheir Binary Images, Finite Fields and their Appl. 17 3, 2011, [10].T. Dougherty, B. Yildiz, and. Karadeniz, Cyclic Codes over R k, Des., Codes, and Cryptog., 63 1, 2012, [11].T. Dougherty, B. Yildiz, and. Karadeniz, elf-dual Codes over R k and Binary elf- Dual Codes, Eur. Journal of Pure and Appl. Math. 6 1, 2013, [12] J. Gao, kew Cyclic Codes over F p + vf p, J. Appl. Math. and Informatics , 2013, [13] M. Rimmer, Isomorphism between kew Polynomial Rings, J. Austral. Math. oc., 25 eries A, 1978, [14] I. iap, T. Abualrub, N. Aydin, and P. eneviratne, kew cyclic codes of arbitrary length, Int. J. Inform. Coding Theory, 2 1, 2011,