Skew-Cyclic Codes over B k

Transcription

1 Skew-Cyclic Codes over B k arxiv: v1 [math.co] 17 May 2017 Irwansyah Mathematics Department, Universitas Mataram, Jl. Majapahit 62, Mataram, INDONESIA Aleams Barra, Intan Muchtadi-Alamsyah, Ahmad Muchlis, Algebra Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, INDONESIA Djoko Suprijanto Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung, 40132, INDONESIA Abstract In this paper we study the structure of θ-cyclic codes over the ring B k including its connection to quasi- θ-cyclic codes over finite field F p r and skew polynomial rings over B k. We also characterize Euclidean self-dual θ-cyclic codes over the rings. Finally, we give the generator polynomial for such codes and some examples of optimal Euclidean θ-cyclic codes. Keywords : θ-cyclic codes, quasi-θ-cyclic codes, the ring B k, Euclidean self-dual codes. 1 Introduction Cyclic codes are some of the most interesting families of codes because of their rich algebraic structures. In 2007, Boucher et. al. [2] introduced a notion of skew-cyclic Contact: djoko@math.itb.ac.id 1

2 codes or θ-cyclic codes as a generalization of cyclic codes. This class of codes has been studied over certain finite rings such as finite fields [2], the ring F 2 +vf 2, where v 2 = v [1], and very recently over the rings A k [7], and produced several optimal codes including optimal Euclidean self-dual code [36,18,11] over F 4 which improve previous known bound for self-dual codes [2]. Moreover, the codes have a close connection with modules over skew-polynomial rings as shown in [1], [2], [3], [6], and [7]. In this paper, we define B k, an algebra over a finite field F p r, which is a natural generalization of the ring A k, and we study its structures including the shape of its maximal ideals and automorphisms. We also study the structures of θ-cyclic codes over the ring B k. We focus on its connection to quasi-θ-cyclic codes over finite fields and skew polynomial rings over B k. We give the generators for such codes. Moreover, we also study self-dual θ-cyclic codes over such rings and give some examples of optimal codes. 2 The ring B k : basic facts In this section we provide some basic facts regarding the ring B k. Some properties are easy to derived, but we include here for the reader s convenience. Let F p r be the field extension of degree r of prime field F p, for some positive integer r. The ring B k is defined as B k := F p r[v 1,v 2,...,v k ]/ v 2 i v i,v i v j v j v i for all i,j = 1,...,k. For example, when k = 1, then B 1 = F p r +vf p r, where v 2 = v. For convenience, we let B 0 = F p r. The ring B k can be considered as a commutative algebra over F p r. Let H be the collection of all subsets of {1,...,k}. Then, we have the following observation. Lemma 1. The ring B k can be viewed as an F p r-vector space with dimension 2 k whose basis consists of elements of the form i H w i, where H H and w i {v i,1 v i } for 1 i k. Proof. As we can see, every element a B k can be written as a = H H α Hv H, for some α H F p r, where v H = i H v i and v = 1. Therefore, B k is a vector space over F p r with basis consists of elements of the form v H = i H v i, where v = 1 and there are k k j=0( j) = 2 k elements of basis. Now, we will show that the set {1,w H2,...,w } is also a basis, where H H2 k 2,...,H 2 k H such that H i H j for all 2 i < j 2 k. Consider, α 1 +α 2 w H2 + +α 2 kw H2 k = 0 for some α i F p r, for all i = 1,...,2 k, which gives, α 1 = α 2 w H2 + +α 2 kw H2 k. If α 1 0, then ξ 1 = ( α 2 w H2 + +α 2 kw H2 k) is a unit, a contradiction to ξ1 w 1,...,w k. So, α 1 = 0, which means, ( α 2 w H2 + +α k+1 w Hk+1 ) = αk+2 w Hk+2 + +α 2 kw H2 k. 2

3 If ( ) α 2 w H2 + +α k+1 w Hk+1 0, then it is a ( contradiction to the fact ) that Hj 2, for all j = k + 2,...,2 k. Consequently, α2 w H2 + +α k+1 w Hk+1 = 0. We have to note that, the set B 1 = {1,w {1},w {1,2},w {2},...,w {1,2,...,k} } is also linearly independent over F p r, because B k is a vector space over F p r with elements of basis are of the form v H, where H H. Therefore, ( ) α 2 w H2 + +α k+1 w Hk+1 = 0 gives α 2 = = α k+1 = 0. On continuing this process, we have α 1 = = α 2 k = 0, which means they are linearly independent over F p r. The following result is an immediate consequence of the above lemma. Lemma 2. The ring B k has characteristic p and cardinality (p r ) 2k. Proof. It is immediate since the characteristic of F p r is p, and B k can be viewed as a F p r-vector space with dimension k i=0( k i) = 2 k. The following lemma gives a characterization for zero divisor elements in B k. Lemma 3. An element ω B k is a zero divisor if and only if ω w 1,w 2,...,w k, where w i {v i,1 v i } for 1 i k. Proof. ( =) It is clear that, v i (1 v i ) = 0, for all i = 1,...,k. Therefore, if ω w 1,w 2,...,w k, then it is a zero divisor in B k. (= ) Consider the equation, (α+βv k )(γ +ǫv k ) = a+bv k given α + βv k,a + bv k B k, for some α,β,a,b B k 1. We have γ = aα 1 and ǫ = (b βa)(α(β +α)) 1. Therefore, if a+bv k = 1, then γ = 1 and ǫ = β(α(β + α)) 1. Which implies, α + βv k is a unit if and only if α and α + β are also units. Considering thisobservationforelements inb k 1,B k 2,...,B 1,wehave α+βv B 1 is a unit if and only if α,α+β F p r are non zero elements. Since, every element in finite commutative ring is either a unit or a zero divisor, we can see that the only zero divisors in B 1 are the elements in the ideals generated by βv or α(1 v). By generalizing this result recursively, we have the intended conclusion. Also, we can easily show that I = w 1,w 2,...,w k is a maximal ideal in B k. Lemma 4. Let I = w 1,w 2,...,w k, where w i {v i,1 v i } for 1 i k. Then I is a maximal ideal in B k. Proof. Consider the quotient ring B k /I. If v i I, then 1 v i 1 mod I, and if 1 v i I, then v i = 1 (1 v i ) 1 mod I. Consequently, B k /I is a field. So, I is a maximal ideal. Moreover, by Lemma 1, every element a in B k, can be written as a = H H α Hw H, for some α H F p r. Therefore, we have B k /I = F p r. Lemma 5. α pr = α, for all α B k. 3

4 Proof. Let α = H H α Hv H, for some α H F p r, where v H = j H v j. Then, for any H 1 H, consider p r ( ) ( ) p p r r i ( ) p r α pr = (α H1 v H1 ) i α H v H = α H1 v H1 + α H v H i i=0 H H 1 H H 1 since F p r has characteristic p and β pr 1 = 1 for all β F p r. If we continue this procedure, then we have α pr = α. The proposition below shows that B k is a principal ideal ring. Proposition 6. Let I = α 1,...,α m be an ideal in B k, for some α 1,...,α m B k. Then, ) pr 1 I = α j. A {1,...,m},A ( 1) A +1( j A ( p Proof. Considerα i A {1,...,m},A ( 1) A +1 j A j) r α 1.ForanyA {1,...,m}, if i A, then ( ) pr 1 α i ( 1) A +1 α j = ( 1) A +1 α i p r 1 α j j A since α pr i {1,...,m}, such that j A {i} = α i by Lemma 5. Consequently, there is a unique A = A {i} α i ( 1) A +1 ( j A α j ) pr 1 +( 1) A +1 ( j A ) pr 1 α j = 0. Otherwise, if i A, then there is a unique A = A {i} {1,...,m} such that ( ) pr 1 ( ) pr 1 α i ( 1) A +1 α j +( 1) A +1 α j = 0. j A So, every term will be vanish except α i α pr i = α i. Therefore, ) pr 1 I α j. It is clear that A {1,...,m},A ( 1) A +1( j A A {1,...,m},A ( 1) A +1( j A Thus, I = A {1,...,m},A ( 1) A +1 ( j A α j) p r 1. j A α j ) pr 1 I. 4

5 The following proposition shows that the ideal in Lemma 4 is the only maximal ideal in B k. Proposition 7. An ideal I in B k is maximal if and only if I = w 1,w 2,...,w k, where w i {v i,1 v i } for 1 i k. Proof. ( =) It is clear by Lemma 4. (= ) Let J be a maximal ideal in B k. By Proposition 6, B k is a principal ideal ring. Then, let J = ω, for some ω B k. Note that, ω is not a unit in B k, so it is a zero divisor. By Lemma 3, ω is an element of some m i = w 1,w 2,...,w k, which means J m i. Consequently, J = m i, because J is a maximal ideal. The following result gives a characterization of automorphism in B k. Theorem 8. Let θ be an endomorphism in B k. Then, θ is an automorphism if and only if θ(v i ) = w j, for every i {1,...,k}, and θ, when restricted to F p r, is an element of Gal(F p r/f p ). Proof. (= ) Let J = v 1,...,v k and J θ = θ(v 1 ),...,θ(v k ). Consider the map λ : B k B k J J θ a+j θ(a)+j θ We can see that the map λ is a ring homomorphism. For any a,b B k /J where λ(a) = λ(b), let a = a 1 + J and b = b 1 + J for some a 1,b 1 B k. As we can see, θ(a 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 B k /J θ, let a = a 2 +J θ for some a 2 B k, then there exists a = θ 1 (a 2 )+J such that λ(a) = a. Therefore, F p r B k /J B k /J θ, which implies J θ is also a maximal ideal. By Proposition 7, J θ = w 1,...,w k, where w i {v i,1 v i } for 1 i k. By Proposition 6, J θ = = A {1,...,k},A ( 1) A +1( j A A {1,...,k},A ( 1) A +1( j A w j ) pr 1 θ(v j ) ) p r 1 which means, and ) pr 1 w j A {1,...,k},A ( 1) A +1( j A A {1,...,k},A ( 1) A +1( j A θ(v j ) ) p r 1 5

6 are associate. Therefore, θ(v i ) = βw j for some unit β which satisfies ( β A ) p r 1 = β, for all A. Since β pr = β and β is a unit, we have β pr 1 = 1. Therefore, β must be equal to 1. Moreover, since θ is an automorphism, θ(v i ) θ(v j ) whenever i j. Also, since all automorphism of F p r are elements of Gal(F p r/f p ), then it is clear that when θ restricted to F p r, it is really an element of Gal(F p r/f p ). ( =) Suppose that θ(v i ) = w j, and θ(v i ) θ(v j ) whenever i j. By Lemma 1, we can see that θ is also an automorphism. 3 Gray map As mentioned implicitly in Lemma 1, every element in the ring B k can be written as α+βv k, where α,β B k 1. For k 2, let φ k : B k B 2 k 1 where φ k (α+βv k ) = (α,α+β). Then define the Gray map Φ k : B k F 2k p r where Φ 1(γ) = φ 1 (γ), Φ 2 (γ) = φ 1 (φ 2 (γ)) and Φ k (α) = 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 it is a generalization of the one in [7]. Also, every element a in B k can be written as a = H Hα H v H for some α H F p r, where v H = i H v i. Now, define a map Ψ k as follows. Ψ k : B k F 2k p r a = ( 2 k i=1 α H i v Hi α H H1 H, α H H2 H,..., α H H2k H). Here, H 1,H 2,...,H 2 k H, where H is a collection of subsets of {1,2,...,k}. We can show that this Ψ k map is also a bijection map and it is a permutation of the Gray map Φ k. Therefore, we can choose all H i such that Ψ k = Φ k. So, from now on we assume that we have already chosen all H i such that Ψ k = Φ k. Let H be an element in H. We shall define a set of automorphisms in the ring B k based on the set H. Define the map Θ i by Θ i (v i ) = v i +1 and Θ i (v j ) = v j j i. For all S H the automorphism Θ S is defined by: Θ S = i S Θ i. 6

7 Note that Θ S is an involution on the ring B k. Moreover, our Θ S here is a generalization of the map Θ S in [7]. Now, let S 1,S 2 be two elements of H with the same cardinality. Let λ S1,S 2 be a one-on-one correspondence between S 1 and S 2 and λ S1,S 2 (i) = i for all i S 1. For any α F p r, we define the map Λ S1,S 2,t as follows. Λ S1,S 2,t(αv i ) = α pt v λs1,s 2 (i), for every i S 1, where 0 t r. Using two class of automorphisms above, we can describe all automorphisms in the ring B k as stated in the following result. Lemma 9. If θ is an automorphism in the ring B k, then there exist S,S 1,S 2, three subsets of {1,...,k}, some integer t, where S 1 = S 2 and 0 t r, such that θ = Θ S Λ S1,S 2,t. Proof. Let φ be the Frobenius automorphism in F p r. If θ Fp r = φ t, then we have t = t. Now, define and S = {j θ 1 (1 v j ) = v i,for some i}, S 1 = {l θ(v l ) = w l,where l l}, S 2 = {s s S 1,such that θ(v s ) = 1 v s }. Consider θ(v i ) for any i in {1,...,k}. We have four cases to consider as listed below. If θ(v i ) = v i, then i is not an element of S, S 1, and S 2. Hence, θ(v i ) = (Θ S Λ S1,S 2,t)(v i ). If θ(v i ) = v j, where j i, then i S 1, but not in S 2 and S. So, Λ S1,S 2,t(v i ) = v j and (Θ S Λ S1,S 2,t)(v i ) = Θ S (v j ) = v j = θ(v i ). If θ(v i ) = 1 v i, then i S 2 S but not in S 1. So, Θ S (v i ) = 1 v i and (Θ S Λ S1,S 2,t)(v i ) = Θ S (v i ) = 1 v i = θ(v i ). If θ(v i ) = 1 v j, where j i, then i S 1 and j S 2 S. We have (Θ S Λ S1,S 2,t)(v i ) = Θ S (v j ) = 1 v j = θ(v i ). Let T be the matrix that performs the cyclic shift on a vector. Let σ i,k be the permutation of {1,2,...,2 k } defined by (σ i,k ) {j2 i +1,...,(j+1)2 i } = T 2i 1 (j2 i +1,...,(j +1)2 i ) for all 0 j 2 k i 1. Let Σ i,k be the permutation on elements of F 2k pr induced by σ i,k. That is, for x = (x 1,x 2,...,x 2 k) F 2k p r, Σ i,k (x) = ( x σi,k (1),x σi,k (2),...,x σi,k (2 k )). (1) Related to the Gray map, we have the following results. 7

8 Lemma 10. Let k 1 and 1 i k. For x B k we have Σ i,k (Φ k (x)) = Φ k (Θ i (x)) Proof. See the proof of [7, Lemma 2.6]. We can extend the definition of Σ i,k to any element of H as follows. Definition 11. For all S H we define the permutation Σ S,k by Σ S,k = Σ i,k. i S It is clear that for all x B k we have Σ S,k (Φ k (x)) = Φ k (Θ S (x)). (2) GivenautomorphismΛ S1,S 2,t,let λ S1,S 2 beapermutationonh i induced byλ S1,S 2, here we assume λ S1,S 2 is a bijection map on {1,...,k}, where λ S1,S 2 (j) = j when j S 1. Then, for any a = H H α Hv H, we have Ψ k (Λ S1,S 2,t(a)) = α pt, H H λ 1 S 1,S 2 (2) α pt H,..., H H λ 1 S 1,S 2 (2 k 1) α pt H, H H 2 k The right hand side of the above equation induced a bijective map Γ S1,S 2,t on F 2k p r which simplify the equation to be the following α pt H. Ψ k Λ S1,S 2,t = Γ S1,S 2,t Ψ k. (3) Furthermore, related to any automorphism in the ring B k, we have the following result. Proposition 12. Let θ be an automorphism in the ring B k. Then, there exist S,S 1,S 2, three subsets of {1,...,k} and some integer t, where S 1 = S 2 and 0 t r, such that Ψ k θ = (Σ S,k Γ S1,S 2,t) Ψ k. Proof. Apply Lemma 9, Lemma 10, and equation (3). 8

9 4 Skew-cyclic codes In this section, we characterize the skew-cyclic codes over the ring B k. We define first the skew-cyclic codes or θ-cyclic codes as a generalization of cyclic codes. Definition 13. Let θ be an automorphism in B k. C Bk n length n over B k if the following two conditions hold: is called θ-cyclic code of (1) C is a B k -submodule of B n k, (2) T θ (c) = (θ(c n 1 ),θ(c 0 ),θ(c 1 ),...,θ(c n 2 )) C, for any c = (c 0,c 1,...,c n 1 ) C. We note that the code C B n k is called linear over B k if C satisfies the property (1) above. We may also generalize the above definition to the quasi-cyclic one. Definition 14. Let θ be an automorphism in F p r. C F n pr is called quasi-θ-cyclic code of length n over F p r of index l if the following two conditions hold: (1) C is an F p r-submodule of F n p r, (2) T l θ (c) = (θ(c n l (mod n)),θ(c n l+1 (mod n) ),θ(c n l+2 (mod n) ),...,θ(c n 1 l (mod n) )) C, for any c = (c 0,c 1,...,c n 1 ) C. (In this case we have l is a divisor of n.) Let S,S 1,S 2 {1,2,...,k}, where S 1 = S 2 and let Ξ S,S1,S 2 = ξ S,S1,S 2,t T 2k be a bijective map on elements of F n2k p where T is the cyclic shift modulo r n2k and ξ S,S1,S 2,t defined for all elements by x = (x 1 1,...,x1 2 k,x 2 1,...,x2 2 k,...,x n 1,...,xn 2 k ) F n2k, 2 ξ S,S1,S 2,t(x) = ξ S,S1,S 2,t((x 1 1,...,x1 2 k,x 2 1,...,x2 2 k...,x n 1,...,xn 2 k )) = ((Σ S,k Γ S1,S 2,t)(x 1 ),(Σ S,k Γ S1,S 2,t)(x 2 ),...,(Σ S,k Γ S1,S 2,t)(x n )) where x j = (x j 1,...,x j 2 k ), for 1 j n. Since T 2k and ξ S,S1,S 2,t commute, Ξ S,S1,S 2,t can be written as T 2k ξ S,S1,S 2,t as well. Now we are ready to provide the first characterization of θ-cyclic codes over B k. Lemma 15 (First characterization). Let C be a code in B n k and θ = Θ S Λ S1,S 2,t be an automorphism in B k, for some S,S 1,S 2 {1,2,...,k} and an integer t, where 0 t r. Then, the code Φ k (C) is fixed by the bijection Ξ S,S1,S 2,t if and only if C is a θ-cyclic code. 9

10 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 p r. We have Ξ S,S1,S 2,t(y) = ξ S,S1,S 2,t T 2k ((Φ k (x 1 ),Φ k (x 2 ),...,Φ k (x n ))) = ξ S,S1,S 2,t((Φ k (x n ),Φ k (x 1 ),...,Φ k (x n 1 ))) = ((Σ S,k Γ S1,S 2,t)(Φ k (x n )),(Σ S,k Γ S1,S 2,t)(Φ k (x 1 )),...,(Σ S,k Γ S1,S 2,t)(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. Let C be a code over F n2k p that is fixed by the permutation Ξ r S,S 1,S 2,t and let x = (x 1,x 2,...,x n ) Φ 1 k (C ). Then, there exists such that y = (x 1 1,...,x2k 1,x1 2,...,x2k 1,...,x1 n,...,x2k n ) C, y = Φ k ((x 1,x 2,...,x n )) = (Φ k (x 1 ),Φ k (x 2 ),...,Φ k (x n )). Since C is fixed by the permutation Ξ S,S1,S 2,t and Φ k (x i ) F 2k pr, we have that for Ξ S,S1,S 2,t(y) C : Ξ S,S1,S 2,t(y) = ξ S,S1,S 2,t T 2k ((Φ k (x 1 ),Φ k (x 2 ),...,Φ k (x n ))) = ξ S,S1,S 2,t((Φ k (x n ),Φ k (x 1 ),...,Φ k (x n 1 ))) = ((Σ S,k Γ S1,S 2,t)(Φ k (x n )),(Σ S,k Γ S1,S 2,t)(Φ k (x 1 )),...,(Σ S,k Γ S1,S 2,t)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 provide the second characterization of θ-cyclic codes over B k. For this purpose, let Ψ k be the map extended from Ψ k, where for any a = (a 1,...,a n ) in B n k, we have Ψ k (a) = (Ψ k (a 1 ),...,Ψ k (a n )). Before providing the second characterization, we need the following lemma. Lemma 16. Let C be a subset of B n k. Then, C is a B k-linear code with length n if and only if there exist linear codes, C 1,...,C 2 k, over F p r such that C = Ψ 1 k (C 1,...,C 2 k). 10

11 Proof. (= )GivenC,thereexist codesc 1,...,C 2 k suchthatc = Ψ 1 k (C 1,...,C 2 k), (as Ψ k is a bijection). For any C i, we want to show that it is really a linear code over F p r. For any c 1,c 2 C i, consider c j = Ψ 1 k (0,...,0,c j,0,...,0) C, for j = 1,2. Since C is linear, we have Ψ k (c 1 + c 2 ) = (0,...,0,c 1 + c 2,0,...,0) (C 1,...,C s ). Also, for any α F p r, there exists α B k such that Ψ k (α ) = (0,...,α,0,...,0) F 2k p r, where α lies in the i-th coordinate. Then, Ψ 1 k (0,...,0,αc 1,0,...,0) = α c 1 C. So, αc 1 C i for any α F p r and c 1 C i. Therefore, C i is a linear code over F p r. ( =)Forany c 1,c 2 C,since C i islinear forall i, wehave Ψ k (c 1 +c 2 ) (C 0,...,c s ). So,c 1 +c 2 C.Also, foranyβ B k,wehaveψ k (βc 1 ) = Ψ k (β)ψ k (c 1 ) (C 1,...,C s ), which means βc 1 C for any β B k and c 1 C, as we hope. Let λ S1,S 2 beapermutationon{1,2,...,2 k }inducedbyθ S Λ S1,S 2,t,andOrd( λ S1,S 2 ) betheorder of λ S1,S 2. Thefollowing theorem alsogives acharacterization forθ-cyclic codes over the ring B k. Theorem 17 (Second characterization). A linear code C over B k is θ-cyclic of length n if and only if there exist quasi- θ-cyclic codes C 1,C 2,...,C 2 k of length n over F p r with index Ord( λ S1,S 2 ), such that C = Ψ 1 k (C 1,C 2,...,C 2 k) where θ = φ tord( λ S1,S 2 ), for some t as in Lemma 9, with φ is the Frobenius automorphism in F p r, and T θ(c i ) C j, where j S S 2, for all i = 1,2,...,2 k. Proof. (= ) By Proposition 16, we can find codes over F p r, C 1,C 2,...,C 2 k, such that, C = Ψ 1 k (C 1,C 2,...,C 2 k). For any c i C i, let c i = (α 1,...,α n ). If, c = Ψ 1 k (0,...,0,c i,0,...,0), then ( ) α 1 v Hi α 1 v H,...,α n v Hi α n v H. H H, H H i H H, H H i So, if we consider Ψ k (T t 1 θ (c)) = (0,...,0,Tt 1 θ (c i),0,...,0), then we have T θ(c i ) is in C j, where j S S 2. By continuing this process, we have T Ord( λ S1,S 2 ) (c θ i ) C i, which means, C i is quasi- θ-cyclic code over F p r with index Ord( λ S1,S 2 ), for all i = 1,...,2 k. ( =) For any c C, we can see that Ψ k (c) (C 1,...,C 2 k). Since C i is quasi- θcyclic code over F p r with index Ord( λ S1,S 2 ), for all i = 1,...,2 k, C 1, and T t 1 θ (C i) C j, where j S S 2, for all i = 1,2,...,2 k, where 1 t 1 2 k. Then we have T θ (c) = Ψ 1 k (T θ(ψ k (c))) C, as we hope. 11

12 Theorem 17 gives us an algorithm to construct skew-cyclic codes over the ring B k as follows. Algorithm 18. Given n, the ring B k, and an automorphism θ. (1) Decompose θ into θ = Θ S Λ S1,S 2,t. (2) Determine Ord( λ S1,S 2 ) and θ = θ Fp r = φ t, where φ is the Frobenius automorphism in F p r. (3) Choose quasi- θ-cyclic codes over F p r, say C 1,...,C 2 k, such that T t 1 θ (C i) C j, where j S S 2, for all i = 1,2,...,2 k. (4) Calculate C = Ψ 1 k (C 1,...,C 2 k). (5) C is a θ-cyclic code over the ring B k. 5 Skew-polynomial rings, self-dual codes, and optimal codes At the end of this section, we construct some optimal self-dual θ-cyclic codes. For the sake of construction, we provide a connection between θ-cyclic codes and polynomial rings over B k. Also, we characterize self-dual θ-cyclic codes. Let θ be anautomorphism in B k. We define a polynomial ring B k [x;θ] with usual addition, and its multiplication defined as ax i bx j = aθ i (b)x i+j. We can see from the above multiplication that B k [x;θ] is not a commutative ring in general. We call this ring skew polynomial ring. Let R = B k [x;θ]/(x n 1) and define a left action of B k [x;θ] on R by h(x) a := h(x) f(x)+(x n 1), for a = f(x)+(x n 1) R and h(x) B k [x;θ]. It is easy to see that this left action is well-defined and R is a left module over B k [x;θ]. Every elements in Bk n could be associated to a polynomial over B k by the following map, τ : Bk n R c = (c 0,c 1,...,c n 1 ) n 1 i=0 c ix i Moreover, τ is a bijection between Bk n and R. Then, we have the following result (c.f. Theorem 17 in [7]). 12

13 Theorem 19. A linear code C over B k is θ-cyclic if and only if τ(c) is a left B k [x;θ]-submodule of B k [x;θ]/(x n 1). For generators of θ-cyclic codes, we have the following result. Proposition20. LetC = Ψ 1 k (C 1,...,C 2 k) be θ-cycliccodesover B k, wherec 1,...,C 2 k are codes over F p r. If C i = g 1i (x),...,g mi (x), for all i = 1,...,2 k, then in some sense. C = g 11 (x),...,g m1 (x),...,g 1s (x),...,g ms (x) Proof. For any c(x) C, there exist c i (x) C i, where 1 i 2 k, such that c(x) = Ψ 1 k (c 1 (x),...,c 2 k(x)). Now, let for all j = 1,...,2 k, then we have as we hope. ( m1 ) c(x) = v H1 α k1 (x)g 1k k=1 m j c j (x) = α kj (x)g jk k=1 m i + +v Hi α ki (x)g ik + +v H2 k k=1 ( m2 k H j H i α k2 k(x)g 2 k k k=1 s j=1 ( mj k=1 ( mj α kj (x)g jk ) )) α kj (x)g jk k=1 For c 1 = (c 10,...,c 1n 1 ),c 2 = (c 20,...,c 2n 1 ) B n k, define n 1 c 1,c 2 = c 1i c 2i which we call Euclidean product. For a code C B n k, we define C = {b B n k b,c = 0, c C}. If C C, then we call C self-orthogonal, and if C = C, then C called Euclidean self-dual. For a linear Euclidean self-dual code over B k, we have the characterization below (follows from [5, Theorem 6.4]). Proposition 21. Let C = Ψ 1 k (C 1,...,C 2 k). C is a self-dual code over B k if and only if C 1,...,C 2 k are also self-dual codes over F p r As an immediate consequence, by applying Theorem 17 and Proposition 21, we have the following result. 13 i=0

14 Theorem 22. A linear code C over B k is Euclidean self-dual θ-cyclic of length n if and only if there exist self-dual quasi- θ-cyclic codes C 1,C 2,...,C 2 k of length n such that C = Ψ 1 k (C 1,C 2,...,C 2 k) where θ = φ t, for some t as in Lemma 9, with φ is the Frobenius automorphism in F p r, and T t 1 θ (C i) for all i = C λt 1 S1,S (i) 1,2,...,2k, where 1 t 1 2 k. 2 The following lemma gives the minimum distance for codes over B k. Lemma 23. Let C be a linear code over B k. If C = Ψ 1 k (C 1,...,C 2 k), for some codes C 1,...,C 2 k over F p r, then d H (C) = min 1 i 2 k d H (C i ). Proof. Suppose min 1 i 2 k d H (C i ) = d H (C j ) for some j, and let c C j such that d H (c) = d H (C j ), then we have So, d H (C) = min 1 i 2 k d H (C i ). d H (Ψ 1 k (0,...,0,c,0,...,0)) = d H (C j ). As an immediate consequence of Lemma 23, we have the following result. Corollary 24. If C = Ψ 1 k (C 1,...,C 2 k) be a θ-cyclic code over B k, where C i are quasi- θ-cyclic code of index Ord( λ S1,S 2 ) over F p r as in Theorem 17, for all i, then d(c) = min{d(c 1 ),...,d(c 2 k)}. This means, we can use the optimal θ-cyclic codes or quasi-θ-cyclic codes over F p r to construct optimal θ-cyclic codes over B k for all k with respect to Hamming weight. In [2], Boucher and Ulmer find some optimal θ-cyclic codes over finite fields, including Euclidean self-dual code[36, 18, 11] which improves previous known bound for self-dual codes of length 36. We use this optimal codes to construct optimal θ- cyclic codes over B k. Some example of optimal Euclidean self-dual codes are shown in Table 1. Here we only use p = 2, r = 2, θ F2 2 is a Frobenius automorphism, and α is a generator of F 2 2. Also, the given generators generate θ-cyclic codes in the way as in the proof of Proposition 20. Acknowledgement D.S. and I are supported by Riset ITB 2016 and Penelitian Unggulan Perguruan Tinggi (ITB-Dikti)

15 References [1] T. Abualrub, N.Aydin, andp. Seneviratne, Onθ-Cyclic Codes over F 2 +vf 2, Australasian Journal of Combinatorics vol. 54, 2012, [2] D. Boucher, and F. Ulmer, Coding with Skew Polynomial Rings, Journal of Symbolic Computation No. 44, 2009, [3] D. Boucher, W. Geiselmann, and F. Ulmer, Skew Cyclic Codes, Appl. Algebra Eng. Commun. Comput. 18, 2007, [4] Y. Cengellenmis, A. Dertli, and S. Dougherty, Codes over an Infinite Family of Rings with a Gray Map, Designs, Codes, and Cryptography, 72(3), 2014, [5] S.T. Dougherty, J.L. Kim, and H. Kulosman, MDS Codes over Finite Principal Ideal Rings, Design, Codes, and Cryptography, 50(1), 2009, Table 1: Table of examples of optimal θ-cyclic codes n d Generator polynomial B k θ 4 3 x 2 +α 2 x+α B 2 v 2 v 1 v 1 v x 6 +x 5 +α 2 x 4 +x 3 +αx 2 +x+1 B 3 v 2 v 3 v 1 v 2 v 3 v x 10 +α 2 x 9 +αx 8 +x 7 +x 6 +x 4 +x 3 +αx 2 +αx+1 x 18 +x 16 +α 2 x 15 +αx 14 +α 2 x 13 +x 12 +αx 10 +αx 9 +αx 8 +α 2 x 6 +x 5 +αx 4 +x 3 +α 2 x 2 +α 2 B 6 x 20 +x 17 +α 2 x 15 +αx 14 +α 2 x 13 +α 2 x 12 +x 11 +x 9 +αx 8 +αx 7 +α 2 x 6 +αx 5 +x 3 +1 v 1 v 2 v B 2 v 1 4 v 3 1 v 4 1 v 4 v 3 B 7 v 1 1 v 2 1 v 2 v 3 v 3 v 1 v 4 v 5 v 5 v 6 v 6 v 5 v 1 1 v 2 1 v 2 v 3 v 3 v 1 v 4 v 5 v 5 v 6 v 6 v 5 v 7 v 7 15

16 [6] J. Gao, Skew Cyclic Codes over F p + vf p, J. Appl. Math. and Informatics 31(3-4), 2013, [7] Irwansyah, A. Barra, S.T. Dougherty, A. Muchlis, I. Muchtadi-Alamsyah, I., P. Solé, D. Suprijanto, D., and O. Yemen, Θ S -Cyclic Codes over A k, International Journal of Computer Mathematics: Computer System Theory, 1(1), 2016,