Linear and Cyclic Codes over direct product of Finite Chain Rings

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1 Proceedings of the 16th International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE July 2016 Linear and Cyclic Codes over direct product of Finite Chain Rings Joaquim Borges 1 Cristina Fernández-Córdoba 1 and Roger Ten-Valls 1 1 Department of Information and Communications Engineering Universitat Autònoma de Barcelona s: joaquimborges@uabcat cristinafernandez@uabcat rogerten@uabcat Abstract We introduce a new type of linear and cyclic codes These codes are defined over a direct product of two finite chain rings The definition of these codes as certain submodules of the direct product of copies of these rings is given and the cyclic property is defined Cyclic codes can be seen as submodules of the direct product of polynomial rings Generator matrices for linear codes and generator polynomials for cyclic codes are determined Key words: Codes over rings linear codes cyclic codes finite chain rings MSC 2000: 94B60 94B25 1 Introduction Linear codes are a special family of codes with rich mathematical structure One of the most studied class of linear codes is the class of linear cyclic codes The algebraic structure of cyclic codes makes easier their implementation For this reason many practically important codes are cyclic The study of codes over rings has been growing since it was proven in [8] that certain notorious non-linear binary codes can be seen as binary images under the Gray map of linear codes over Z 4 In particular the family of codes over chain rings has received much attention because it includes some good codes (eg [6] [9]) In recent times linear codes with sets of coordinates over different rings are studied (eg Z α 2 Zβ 4 in [3] Zα p r Zβ ps in [2]) Also linear cyclic codes over these kind of structures are studied see [1] [4] [5] and [7]

2 Linear and Cyclic Codes over direct product of Finite Chain Rings In this paper we present the structure of linear and cyclic codes over direct product of finite chain rings R 1 and R 2 Linear codes can be seen as certain submodules of R α 1 Rβ 2 and cyclic codes as submodules of R αβ = R 1[x] x α 1 R 2[x] We determine the generator x β 1 matrix in standard form and the generator polynomials in the cyclic case Finally we present examples to illustrate some particular cases 2 Review of cyclic codes over finite chain rings Let R be a finite chain ring with maximal ideal γ and let e be the nilpotency of γ It is well-know that there exist a prime p and a positive integer m such that R/ γ = q = p m and R = q e = p me Let C be a cyclic code of length n over R It is known that we can identify C as an ideal of R[x]/(x n 1) We assume that n is a positive integer such that it is coprime with p Therefore the polynomial x n 1 has a unique decomposition as a product of basic irreducible polynomials that are pairwise coprime over R[x] Theorem 21 ([6 Theorem 35]) Let C be a cyclic code of length n over a finite chain ring R which has maximal ideal γ and e is the nilpotency of γ Then there exist polynomials g 0 g 1 g e 1 in R[x] such that C = g 0 γg 1 γ e 1 g e 1 and g e 1 g e 2 g 1 g 0 (x n 1) Let C = g 0 γg 1 γ e 1 g e 1 be a cyclic code of length n and let g = g 0 + γg γ e 1 g e 1 Since g 0 is a factor of x n 1 and for i = 1 e 1 the polynomial g i is a factor of g i 1 we will denote ĝ 0 = xn 1 g 0 and ĝ i = g i 1 g i for i = 1 e 1 Define G = e 1 i=0 ĝi then ) it is clear that Gg = g = 0 over R[x]/(x n 1) ( e 1 i=0 ĝi Lemma 22 Let C be a cyclic code of length n over a finite chain ring R which has maximal ideal γ and e is the nilpotency of γ Let g 0 g 1 g e 1 in R[x] such that C = g 0 γg 1 γ e 1 g e 1 and g e 1 g e 2 g 1 g 0 (x n 1) Then 1 γ e 1 g = γ e 1 g e 1 G ĝ 0 2 γ e 1 i ( i 1 j=0 ĝj)g = γ e 1 g e 1 G ĝ i for i = 1 e 1 Proof Let g = g 0 + γg γ e 1 g e 1 Then γ e 1 g = γ e 1 g 0 1 g 1 g 1 g 2 g e 3 g e 2 g e 2 g e 1 g e 1 = γ e 1 g e 1 ĝ 1 ĝ 2 ĝ e 2 ĝ e 1 = γ e 1 g e 1 G ĝ 0

3 J Borges C Fernández-Córdoba and R Ten-Valls and 1 holds For i = 1 e 1 we have that i 1 γ e 1 i ( j=0 and statement 2 is proved i 1 ĝ j )g =γ e 1 i ( j=0 ĝ j )γ i g i 1 g i+1 g i+1 g i+2 g e 2 g e 1 g e 1 =γ e 1 g e 1 ĝ 0 ĝ 1 ĝ i 1 ĝ i+1 ĝ e 1 = γ e 1 g e 1 G ĝ i Corollary 23 (of Th 21) Let C be a cyclic code of length n over a finite chain ring R such that C = g 0 γg 1 γ e 1 g e 1 with g e 1 g e 2 g 1 g 0 (x n 1) Then C = R/ γ e 1 i=0 (e i) deg(ĝ i) Proof From the previous definition of ĝ i these polynomials are the same polynomials described in [6 Theorem 34] Theorem 24 Let C be a cyclic code of length n over a finite chain ring R which has maximal ideal γ and e is the nilpotency of γ Let g 0 g 1 g e 1 polynomials in R[x] such that C = g 0 γg 1 γ e 1 g e 1 and g e 1 g e 2 g 1 g 0 (x n 1) Then the polynomial g = g 0 + γg γ e 1 g e 1 is a generating polynomial of C ie C = g Proof Clearly g = g 0 +γg 1 + +γ e 1 g e 1 C then we only have to prove that γ i g i g for all i = 0 e 1 g Case e = 2: We have that g = g 0 + γg 1 and C = g 0 γg 1 Then γg = γg 0 = γg 0 1 g 1 and xn 1 x g 0 g = γg n 1 1 g 0 since g 0 g 1 and xn 1 g 0 are coprime then γg 1 belongs to g and hence g 0 also belongs to g Case e = 3: We have that g = g 0 + γg 1 + γ 2 g 2 and C = g 0 γg 1 γ 2 g 2 Now γ 2 g = γ 2 g g 1 g 0 2 g 2 g 1 γ xn 1 g 0 g = γ 2 x g n 1 g 1 2 g 0 g 2 and xn 1 g 0 g 0 g 1 g = γ 2 x g n 1 g 0 2 g 0 g 1 since gcd( g 0 g 1 g 1 g 2 xn 1 g 0 ) = 1 then γ 2 g 2 belongs to g hence g 0 + γg 1 So g = g 0 + γg 1 γ 2 g 2 Arguing as in case e = 2 it is straightforward that g = g 0 γg 1 γ 2 g 2 In the general case let g = g 0 + γg γ e 1 g e 1 and define as in Lemma 22 the polynomials G and ĝ i for i {0 e 1} g and γ e 1 i ( i 1 j=0 ĝj)g = γ e 1 G g e 1 ĝ i g for i {1 e 1} Since gcd( G ĝ 0 G G ĝ 1 ĝ e 1 ) = 1 we have that γ e 1 g e 1 g and g = g 0 + γg γ e 2 g e 2 γ e 1 g e 1 Reasoned similarly one obtains that g = g 0 γg 1 γ e 1 g e 1 Then γ e 1 g = γ e 1 g e 1 G ĝ 0 Theorem 25 Let C = g = g 0 + γg γ e 2 g e 2 + γ e 1 g e 1 be a cyclic code of length n over a finite chain ring R which has maximal ideal γ and e is the nilpotency of

4 Linear and Cyclic Codes over direct product of Finite Chain Rings γ with g e 1 g e 2 g 1 g 0 (x n 1) We define the sets [ ] j 1 deg(ĝj ) S γ j = x i ( ĝ t )g for 0 j < e Then S = e 1 j=0 S γ j i=0 forms a minimal generating set for C as an R-module Proof Let c C Then c = dg with d R[x] If deg(d) < deg(ĝ 0 ) then dg S γ 0 R and c S R Otherwise compute d = d 0 ĝ 0 + r 0 with deg(r 0 ) < deg(ĝ 0 ) so dg = d 0 ĝ 0 g + r 0 g and r 0 g S γ 0 R If deg(d 0 ) < deg(ĝ 1 ) then d 0 ĝ 0 g S γ 1 R and c S R Otherwise compute d 0 = d 1 ĝ 1 + r 1 with deg(r 1 ) < deg(ĝ 1 ) so d 0 ĝ 0 g = d 1 ĝ 1 ĝ 0 g + r 1 ĝ 0 g and r 1 ĝ 0 g S γ 1 R In the worst case and reasoning similarly one obtains that c S R if d e 2 ( e 2 ĝt)g S R It is obvious that if deg(d e 2 ) < deg(ĝ e 1 ) then d e 2 ( e 2 ĝt)g S γ e 1 R if not d e 2 = d e 1 ĝ e 1 + r e 1 Therefore e 2 d e 2 ( e 1 ĝ t )g = d e 1 ( e 2 ĝ t )g + r e 1 ( e 2 ĝ t )g = r e 1 ( ĝ t )g S γ e 1 Since r e 1 ( e 2 ĝt)g S γ e 1 then c S R so S is a generating set By the definition of S clearly S = R/ γ e 1 i=0 (e i) deg(ĝi) By Corollary 23 C = S and S is a minimal generating set 3 Linear codes over R α 1 R β 2 Let R 1 and R 2 be finite chain rings where γ 1 and γ 2 are generators of the maximal ideals of R 1 and R 2 with nilpotency indices e 1 and e 2 respectively We will suppose that R 1 and R 2 have the same residue field K = R 1 / γ 1 = R 2 / γ 2 with K = q = p m By : R i K we will denote the natural projection that maps r r = r + γ i for i = 1 or 2 Let T 1 = {r 0 r q 1 } and T 2 = {r 0 r q 1 } be the Teichmüller sets of representatives of R 1 and R 2 resp then we can arrange the subscripts such that r i = r i Assume that e 1 e 2 Then we can consider the surjective ring homomorphism π : R 2 R 1 γ 2 γ 1 r j r j

5 J Borges C Fernández-Córdoba and R Ten-Valls Note that π(γ i 2 ) = 0 if i e 1 For a R 2 and b R 1 define a multiplication as follows: a b = π(a)b Then R 1 is an R 2 -module with external multiplication given by π Since R 1 is commutative then has the commutative property Then we can generalize this multiplication over the ring R α 1 Rβ 2 as follows Let a be an element of R 2 and u = (u u ) = (u 0 u 1 u α 1 u 0 u 1 u β 1 ) Rα 1 Rβ 2 Then a u = (π(a)u 0 π(a)u 1 π(a)u α 1 au 0 au 1 au β 1 ) With this external operation the ring R α 1 Rβ 2 is also an R 2-module Definition 31 A subset C R α 1 Rβ 2 is a linear code if it is a submodule of Rα 1 Rβ 2 The next result gives the structure of a generator matrix of a linear code Proposition 32 Let C be a linear code over R α 1 Rβ 2 Then C is permutation equivalent to a code with generator matrix of the form ( ) B T G = S A where B = I k0 B 01 B 02 B 03 B 0e1 1 B 0e1 0 γ 1 I k1 γ 1 B 12 γ 1 B 13 γ 1 B 1e1 1 γ 1 B 1e1 0 0 γ 2 1 I k 2 γ 2 1 B 23 γ 2 1 B 2e 1 1 γ 2 1 B 2e 1 T = A = S = γ e I ke1 1 γe B e1 1e 1 0 γ e 2 e 1 2 T 01 γ e 2 e 1 2 T 02 γ e 2 e 1 2 T 0e1 0 0 γ e 2 e T 12 γ e 2 e T 1e γ e T e1 1e 1 I l0 A 01 A 02 A 03 A 0e1 1 A 0e2 0 γ 2 I l1 γ 2 A 12 γ 2 A 13 γ 2 A 1e2 1 γ 2 A 1e2 0 0 γ 2 2 I l 2 γ 2 2 A 23 γ 2 2 A 2e 2 1 γ 2 2 A 2e γ e I le2 1 γe A e2 1e 2 0 S 01 S 02 S 0e1 1 S 0e1 0 S 11 S 12 S 1e1 1 S 1e1 0 S e2 e 1 11 S e2 e 1 12 S e2 e 1 1e 1 1 S e2 e 1 1e γ 1 S e2 e 1 2 γ 1 S e2 e 1 e 1 1 γ 1 S e2 e 1 e γ e S e2 3e 1 1 γ e S e2 2e γ e S e2 2e where the entries in γ i 1 B ij and γ i 1 S ij are in γ i 1 and the ones in γt 2 A tj and γ t 2 T tj are in γ t 2

6 Linear and Cyclic Codes over direct product of Finite Chain Rings Proof Similiar to [2 Theorem 4] Let C X be the canonical projection of C on the first α coordinates and C Y on the last β coordinates The canonical projection is a linear map Then C X and C Y are R 1 and R 2 linear codes of length α and β respectively A code C is called separable if C is the direct product of C X and C Y ie C = C X C Y Moreover if C is separable then ( ) B 0 G = 0 A Example 1 Let C be a linear code over Z 3 2 ( Z2 [u] u 3 ) 4 generated by the matrix u u + u u 1 + u u u u 2 0 u u 2 u u + u 2 0 Hence as described in Theorem 32 C is permutation equivalent to a code generated by the following matrix: G = u u u + u 2 Note that C is not separable 4 Cyclic codes over R α 1 R β 2 Definition 41 Let C be a linear code over R α 1 Rβ 2 The code C is called cyclic if (u 0 u 1 u α 2 u α 1 u 0 u 1 u β 2 u β 1 ) C implies (u α 1 u 0 u 1 u α 2 u β 1 u 0 u 1 u β 2 ) C Let u = (u 0 u 1 u α 1 u 0 u β 1 ) be a codeword in C and let i be an integer We then denote by u (i) = (u 0+i u 1+i u α 1+i u 0+i u β 1+i ) the ith shift of u where the subscripts are read modulo α and β respectively We remark that in this paper the definition of a cyclic code over R α 1 Rβ 2 is clear as long as R 1 and R 2 are different rings since the elements on the first α coordinates and the ones in the last β coordinates belong from different rings R 1 and R 2 respectively In the

7 J Borges C Fernández-Córdoba and R Ten-Valls particular case that R 1 = R 2 the cyclic code over R α 1 Rβ 2 is known in the literature as double cyclic code see [4] [7] The term double cyclic is given in order to distinguish the cyclic code over R α 1 Rβ 1 from the cyclic code over Rα+β 1 Note that C X and C Y are R 1 and R 2 cyclic codes of length α and β respectively Denote by R αβ the ring R 1 [x]/(x α 1) R 2 [x]/(x β 1) There is a bijective map between R α 1 Rβ 2 and R αβ where u = (u 0 u 1 u α 1 u 0 u β 1 ) maps to u(x) = (u 0 + u 1 x + + u α 1 x α 1 u u β 1 xβ 1 ) Note that we can extend the map π to the polynomial ring R 2 [x] applying this map to each of the coefficients of a given polynomial Definition 42 Define the operation : R 2 [x] R αβ R αβ as λ(x) (p(x) q(x)) = (π(λ(x))p(x) λ(x)q(x)) where λ(x) R 2 [x] and (p(x) q(x)) R αβ The ring R αβ with the external operation is a R 2 [x]-module Let u(x) = (u(x) u (x)) be an element of R αβ Note that if we operate u(x) by x we get x u(x) = x (u(x) u (x)) = (u 0 x + + u α 1 x α u 0x + + u β 1 xβ ) = (u α u α 2 x α 1 u β u β 2 xβ 1 ) Hence x u(x) is the image of the vector u (1) Thus the operation of u(x) by x in R αβ corresponds to a shift of u In general x i u(x) = u (i) (x) for all i 41 Algebraic structure and generators of cyclic codes In this subsection we study submodules of R αβ We describe the generators of such submodules and state some properties From now on S will denote the R 2 [x]-submodule generated by a subset S of R αβ For the rest of the discussion we will consider that α and β are coprime integers with p From this assumption we know that R 1 [x]/ x α 1 and R 2 [x]/ x β 1 are principal ideal rings see [6] Theorem 43 Every submodule C of the R 2 [x]-module R αβ can be written as C = (b(x) 0) (l(x) a(x)) where b(x) a(x) are generator polynomials in R 1 [x]/(x α 1) and R 2 [x]/(x β 1) resp and l(x) R 1 [x]/(x α 1)

8 Linear and Cyclic Codes over direct product of Finite Chain Rings Proof Let ψ X : R αβ R 1 [x]/(x α 1) and ψ Y : R αβ R 2 [x]/(x β 1) be the canonical projections let C be a submodule of R αβ Define C = {(p(x) q(x)) C q(x) = 0} It is easy to check that C = ψ X (C ) by (p(x) 0) p(x) Hence by Theorem 24 ψ X (C ) is finitely generated by one element and so is C Let b(x) be a generator of ψ X (C ) then (b(x) 0) is a generator of C As R 2 [x]/(x β 1) is also a principal ideal ring then C Y = ψ Y (C) is generated by one element Let a(x) C Y such that C Y = a(x) then there exists l(x) R 1 [x]/(x α 1) such that (l(x) a(x)) C We claim that C = (b(x) 0) (l(x) a(x)) Let (p(x) q(x)) C then q(x) = ψ Y (p(x) q(x)) C Y So there exists λ(x) R 2 [x] such that q(x) = λ(x)a(x) Now (p(x) q(x)) λ(x) (l(x) a(x)) = (p(x) π(λ(x))l(x) 0) belongs to C Then there exists µ(x) R 2 [x] such that (p(x) π(λ(x))l(x) 0) = µ(x) (b(x) 0) Thus (p(x) q(x)) = µ(x) (b(x) 0) + λ(x) (l(x) a(x)) So C is finitely generated by (b(x) 0) (l(x) a(x)) From the previous results it is clear that we can identify double cyclic codes in R α 1 Rβ 2 as submodules of R αβ So any submodule of R αβ is a cyclic code From now on we will denote by C indistinctly both the code and the corresponding submodule of R αβ In the following a polynomial f(x) in R 1 [x] or R 2 [x] will be denoted simply by f Proposition 44 Let C be a cyclic code over R α 1 Rβ 2 Then there exist polynomials l and b e1 1 b e1 2 b 1 b 0 (x α 1) over R 1 [x] and a e2 1 a e2 2 a 1 a 0 (x β 1) over R 2 [x] such that C = (b 0 + γ 1 b γ 1 e 1 1 b e1 1 0) (l a 0 + γ 2 a γ 2 e 2 1 a e2 1) Proof Let C be a cyclic code over R α 1 Rβ 2 By Theorem 43 there exist polynomials b l R 1 [x]/(x α 1) and a R 2 [x]/(x β 1) such that C = (b 0) (l a) By Theorem 24 one can consider b = b 0 + γ 1 b γ e b e1 1 and a = a 0 + γ 2 a γ e a e2 1 such that b e1 1 b e1 2 b 1 b 0 (x α 1) and a e2 1 a e2 2 a 1 a 0 (x β 1) For the rest of the discussion any cyclic code C over R α 1 Rβ 2 is of the form C = (b 0) (l a) where b = b 0 + γ 1 b γ e b e1 1 and a(x) = a 0 + γ 2 a γ e a e2 1 for polynomials b i and a j as in Proposition 44

9 J Borges C Fernández-Córdoba and R Ten-Valls Example 2 Let R 1 = F 9[u] u 2 and R 2 = F 9[u] u 3 with γ 1 = u e 1 = 2 γ 2 = u and e 2 = 3 Let ξ be a generator of the multiplicative group F 9 Consider the cyclic code over ( F9 [u] u 2 ) 4 ( F9 [u] u 3 ) 10 generated by Then C = (u(x 2 1) 0) (u (x 4 + ξ 3 x 2 + 1) + u(x 2 + ξ 5 x + 1) + u 2 ) b 0 = x 4 1 b 1 = x 2 1 l = u a 0 = x 4 + ξ 3 x a 1 = x 2 + ξ 5 x + 1 a 2 = 1 42 Minimal generating sets Our goal is to find a set generators for a cyclic code C as an R 2 -module Once we found it we are going to use it to determine the size of C in terms of the generator polynomials Since b 0 is a factor of x α 1 and for i = 1 e 1 1 the polynomial b i is a factor of b i 1 we will denote ˆb 0 = xα 1 b 0 we define â 0 = xβ 1 a 0 ˆb i = b i 1 b i for i = 1 e 1 1 and ˆb e1 = b e1 1 In the same way â j = a j 1 a j for j = 1 e 2 1 and â e2 = a e2 1 Theorem 45 Let C be a cyclic code over R α 1 Rβ 2 which has maximals ideals γ 1 R 1 and γ 2 R 2 with nilpotent indices e 1 and e 2 respectively Define for 0 j < e 1 and for 0 k < e 2 Then [ j 1 B j = x i ( ˆbt ) (b 0) [ k 1 A k = x i ( â t ) (l a) S = e 1 1 B j j=0 ] deg(ˆbj ) 1 i=0 ] deg(âk ) 1 i=0 ( ) e 2 1 A k forms a minimal generating set for C as an R 2 -module Moreover where q is the cardinality of the residue field k=0 e1 1 C = q i=0 (e 1 i) deg(ˆb i )+ e 2 1 j=0 (e 2 j) deg â j

10 Linear and Cyclic Codes over direct product of Finite Chain Rings Proof ( By Theorem 25 it is clear that the elements in S are R 2 -lineal independent since e1 ) ( 1 j=0 B j and e2 ) 1 k=0 A k are minimal generating sets for the codes C X and C Y X Y respectively Let c be a codeword of C then c = q (b 0) + d (l a) Reasoning similarly as in Theorem 25 we have that q (b 0) e 1 1 j=0 B j R2 So we have to prove that d (l a) S R2 If deg(d) < deg(â 0 ) then d (l a) A 0 R2 and c S R2 Otherwise compute d = d 0 â 0 + r 0 with deg(r 0 ) < deg(â 0 ) so d (l a) = d 0 â 0 (l a) + r 0 (l a) and r 0 (l a) A 0 R2 In the worst case and reasoning similarly one obtains that c S R2 if d e2 2( e 2 2 ât) (l a) S R2 It is obvious that if deg(d e2 2) < deg(â e2 1) then d e2 2( e 2 2 ât) (l a) A e2 1 R2 if not d e2 2 = d e2 1â e2 1 + r e2 1 Therefore ( e2 ) ( 2 e2 ) ( 1 e2 ) 2 d e2 2 â t (l a) = d e2 1 â t (l a) + r e2 1 â t (l a) On one hand r e2 1( e 2 2 ât) (l a) A e2 1 R2 On the other hand d e2 1( e 2 1 ât) (l a) = d e2 1( e 2 1 ât) (l 0) and then d e2 1( e 2 1 ât) (l a) = f (b 0) e 1 1 j=0 B j R2 Thus c S R2 and S is a minimal generating set for C Example 3 Consider R αβ = Z 2 [x]/(x 2 1) Z 4 [x]/(x 3 1) and the cyclic code C = (x 1 (x 2 + x + 1) + 2) where b 0 (x) = x 2 1 l(x) = x 1 a 0 (x) = x 2 + x + 1 and a 1 (x) = 1 Then S = {(x 1 x 2 + x + 3) (0 2x + 2) (0 2x 2 + 2x)} and 2 1 C = 2 j=0 (2 j)deg(â j) = 2 4 = 16 Example 4 From Example 2 consider the cyclic code over by ( F9 [u] u 2 C = (u(x 2 1) 0) (u (x 4 + ξ 3 x 2 + 1) + u(x 2 + ξ 7 x + 1) + u 2 ) ) 4 ( F9 [u] u 3 ) 10 generated Let b = u(x 2 1) l = u and a = (x 4 + ξ 3 x 2 + 1) + u(x 2 + ξ 7 x + 1) + u 2 Then a minimal generating set for C is the union of and B 0 = B 1 = [ x i (u(x 2 1) 0) ] 1 i=0 A 0 = [ x i (l a) ] 5 i=0 A 1 = [ x i µ (l a) ] 1 i=0 A 2 = [ x i λ (l a) ] 1 i=0 where µ = x 6 + ξ 7 x 4 + ξ 3 x and λ = x 8 + ξx 7 + ξx 6 + x 5 + 2x 3 + ξ 5 x 2 + ξ 5 x + 2 So C = i=0 (2 i)deg(ˆb i )+ 3 1 j=0 (3 j)deg(â j) = = 9 26

11 J Borges C Fernández-Córdoba and R Ten-Valls Acknowledgements This work has been partially supported by the Spanish MINECO grants TIN P and MTM REDT and by the Catalan AGAUR grant 2014SGR-691 References [1] T Abualrub I Siap N Aydin Z 2 Z 4 -additive cyclic codes IEEE Trans Info Theory 60(3) (2014) [2] I Aydogdu I Siap On Z p rz p s-additive codes Linear and Multilinear Algebra 63(10) (2014) [3] J Borges C Fernández-Córdoba J Pujol J Rifà and M Villanueva Z 2 Z 4 -linear codes: generator matrices and duality Des Codes and Crypto 54(2) (2010) [4] J Borges C Fernández-Córdoba R Ten-Valls Z 2 -double cyclic codes arxiv: [5] J Borges C Fernández-Córdoba R Ten-Valls Z 2 Z 4 -additive cyclic codes generator polynomials and dual codes arxiv: [6] H Q Dinh S R López-Permouth Cyclic and negacyclic codes over finite chain rings IEEE Trans Info Theory 50(8) (2004) [7] J Gao M Shi T Wu and F Fu On double cyclic codes over Z 4 Finite Fields and Their Applications 39 (2016) [8] A R Hammons P V Kumar A R Calderbank N J A Sloane P Solé The Z 4 -Linearity of Kerdock Preparata Goethals and Related Codes IEEE Trans Info Theory 40(2) (1994) [9] G Norton A Salagean On the structure of linear and cyclic codes over finite chain rings Appl Algebra Eng Commun Comput 10 (2000)

Linear, Cyclic and Constacyclic Codes over S 4 = F 2 + uf 2 + u 2 F 2 + u 3 F 2

Filomat 28:5 (2014), 897 906 DOI 10.2298/FIL1405897O Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Linear, Cyclic and Constacyclic