The Structure of Z [u]Z [u, v]-additive codes ∗ 2 2 N. Annamalai Research Scholar Department of Mathematics 6 Bharathidasan University 1 0 Tiruchirappalli-620 024, Tamil Nadu, India 2 n Email: algebra.annamalai@gmail.com a J 9 C. Durairajan 1 Assistant Professor ] T Department of Mathematics I . s c School of Mathematical Sciences [ Bharathidasan University 1 v Tiruchirappalli-620024, Tamil Nadu, India 9 5 Email: cdurai66@rediffmail.com 8 4 0 . 1 0 6 1 : v Proposed running head: The Structure of Z [u]Z [u,v]-additive codes i 2 2 X r a ∗ The first author would like to thank the Department of Science and Technology (DST), New Delhi, India for their financial support in the form of INSPIRE Fellowship (DST Award Letter No.IF130493/DST/INSPIREFellowship / 2013/(362)Dated: 26.07.2013)to carry out this work. 1 Abstract In this paper, we study the algebraic structure of Z [u]Z [u,v]-additive codes 2 2 which are Z [u,v]-submodules where u2 = v2 = 0 and uv = vu. In particular, we 2 determineaGraymapfromZ [u]Z [u,v]toZ2α+8β andstudygeneratorandparity 2 2 2 check matrices for these codes. Further we study the structure of Z [u]Z [u,v]- 2 2 additive cyclic codes and constacyclic codes. Keywords: Additive codes; Generator Matrix; Parity-check Matrix. 2000 Mathematical Subject Classification: Primary: 94B25, Secondary: 11H31 Corresponding author: Dr. C. Durairajan Assistant Professor Department of Mathematics Bharathidasan University Tiruchirappalli-620024, Tamil Nadu, India E-mail: cdurai66@rediffmail.com 2 1 Introduction Let Z be the ring of integers modulo 2. Let Zn denote the set of all binary vectors of 2 2 lenght n. Any non-empty subset of Zn is a binary code of length n and a subgroup of Zn 2 2 is called a binary linear code. In this paper, we introduce a subfamily of binary linear codes, called Z [u]Z [u,v]-linear codes. 2 2 Additive codes were first defined by Delsarte in 1973 in terms of association schemes [5], [9]. In a translation association scheme, an additive code is generally defined as a subgroup of the underlying abelian group. Codes over finite rings were studied in [6], [8] and with the remarkable paper by Hammons et.al[5] have been studied intensively for the last four decades. A study on mixed alphabet codes has been introduced and some bounds have been presented by Brouwer et.al.[1]. Z Z Z Z Later, -additive codes were generalized to -additive codes by Aydogu and 2 4 2 2s Siap [2]. For s ≥ 2, these generalisations to Z Z -additive codes are interesting since 2 2s they provided good binary codes via Gray maps with rich algebraic structure. Further- Z Z more, the structure of -additive codes and their duals has been determined [3]. pr ps In this correspondance, begin inspired by these additive codes, a generalization towards another direction that have a good algebraic structure and provide good binary codes is presented. The aim of this paper is the study of the algebraic structure of Z [u]Z [u,v]-additive 2 2 codes and their dual codes. It is organized as follows. In Section II, we recall the linear codes over the ring Z + uZ and the linear codes over the ring Z + uZ + vZ + 2 2 2 2 2 uvZ . In Section III, we study the concept of Z [u]Z [u,v]-additive codes and duality, 2 2 2 we determine the generator matrices of the these codes. In section IV, we study the structure of Z [u]Z [u,v]-additive cyclic codes. In section V, we study the structure of 2 2 Z [u]Z [u,v]-additive constacyclic codes. 2 2 2 Linear Codes over Chain Rings Let R be a ring, then Rn is an R-module. A submodule of Rn is called R-linear code. We denote the Hamming weight of a word as w and Lee weight as w . H L The set Z +uZ = {0,1,u,1+u}, where u2 = 0, is a commutative finite chain ring 2 2 of four elements. We denote this ring as R . 1 In [7], a R -linear code C is permutation equivalent to a code with generator matrix 1 I A B +uB G = l0 1 2 , " 0 uIl1 uD # 3 where A,B ,B and D are matrices over Z . 1 2 2 Define w (a+ ub) = w (b,a⊕ b) for all a,b ∈ Z where the Hamming weight of a L H 2 word is the number of non-zero coordinates in it. The definition of the weight immediately leads to a Gray map from R to Z2 which 1 2 can naturally be extended to Rn: 1 χ : (Rn,Lee weight) → (Z2n,Hamming weight) is defined as: L 1 2 χ (a¯+u¯b) = (¯b,a¯⊕¯b), L where a¯,¯b ∈ Zn and ⊕ denotes addition in Z . 2 2 Let R be the ring Z [u,v] := Z + uZ + vZ + uvZ with u2 = 0, v2 = 0 and 2 2 2 2 2 2 uv = vu where Z = {0,1}. Then R is a commutative finite chain ring of 16 elements 2 2 and a+ub+vc+uvd ∈ R is a unit iff a 6= 0. 2 The group of units of R is given by R∗ = {1,1+u,1+v,1+uv,1+u+v,1+u+ 2 2 v + uv,,1+ u + uv,1 + v + uv}. It is also a local ring with the unique maximal ideal uvR . 2 A linear code C of lenght n over the ring R is an R -submodule of Rn. 2 2 2 A non-zero R -linear code C has a generator matrix which after a suitable permuta- 2 tion of the coordinates can be written in the form: I A A A A k0 01 02 03 04 0 uI uA uA uA G =  k1 12 13 14  (2.1) 0 0 vI vA vA k2 23 24    0 0 0 uvI uvA   k3 34   where A are all matrices over Z . ij 2 To define the Lee weights and Gray maps for codes over R . Define w (a + ub + 2 L vc+uvd) = w (d,a⊕d,b⊕d,a⊕b⊕d,c⊕d,a⊕c⊕d,b⊕c⊕d,a⊕b⊕c⊕d) for all H a,b,c,d ∈ Z . The definition of the weight immediately leads to a Gray map from R to 2 2 Z8 which can naturally be extended to Rn: 2 φ : (Rn,Lee weight) → (Z8n,Hamming weight) L 2 2 is defined as: φ (a¯+u¯b+vc¯+uvd¯) = (d¯,¯a⊕d¯,¯b⊕d¯,¯a⊕¯b⊕d¯,c¯⊕d¯, L a¯⊕c¯⊕d¯,¯b⊕c¯⊕d¯,a¯⊕¯b⊕c¯⊕d¯) where a¯,¯b,c¯,d¯∈ Zn. 2 Z Z 3 [u] [u,v]-additive codes 2 2 We know that the ring Z [u] is a subring of the ring R . Being inspired by the structure 2 2 of Z Z [u]-additive codes, we define the following set: 2 2 4 Z [u]Z [u,v] = {(a,b) | a ∈ Z [u] and b ∈ Z [u,v]}. 2 2 2 2 The set Z [u]Z [u,v] cannot be endowed with algebraic structure directly. It is not 2 2 well defined with respect to the usual scalar multiplication by v ∈ R . Therefore, this 2 set is not an R -module. To make it well defined and enrich with an algebraic structure 2 we introduce a new scalar multiplication as follows: We define a mapping η : R → Z [u], 2 2 η(a+ub+vc+uvd) = a+ub i.e., η(x) = x mod v. It is easy to conclude that η is a ring homomorphism. Using this map, we define a scalar multiplication. For v = (a0,a1,··· ,aα−1,b0,b1,··· ,bβ−1) ∈ Z2[u]α ×R2β and l ∈ R2, we have lv = (η(l)a0,η(l)a1,··· ,η(l)aα−1,lb0,lb1,··· ,lbβ−1). (3.1) A non-empty subset C of Rα×Rβ is called Z [u]Z [u,v]-additive code if it is a R - 1 2 2 2 2 submodule of Rα×Rβ with respect to the scalar multiplication defined in Eq.3.1. Then 1 2 the binary image Φ(C) = D is called Z [u]Z [u,v]-linear code of length n = 2α + 8β 2 2 where Φ is a map from Rα ×Rβ to Zn defined as 1 2 2 Φ(x,y) = (q0,q1,··· ,qα−1,q0 +r0,··· ,qα−1 +rα−1, d0,d1,··· ,dβ−1,a0 +d0,a1 +d1,··· ,aβ−1 +dβ−1, b0 +d0,b1 +d1,··· ,bβ−1 +dβ−1, a0 +b0 +d0,a1 +b1 +d1,··· ,aβ−1 +bβ−1 +dβ−1, c0 +d0,c1 +d1,··· ,cβ−1 +dβ−1, a0 +c0 +d0,a1 +c1 +d1,··· ,aβ−1 +cβ−1 +dβ−1, b0 +c0 +d0,b1 +c1 +d1,··· ,bβ−1 +cβ−1 +dβ−1, a0 +b0 +c0 +d0,··· ,aβ−1 +bβ−1 +cβ−1 +dβ−1), for all x = (x0,x1,··· ,xα−1) ∈ R1α where xi = ri + uqi for 0 ≤ i ≤ α − 1 and y = (y0,y1,··· ,yβ−1) ∈ R2β, where yj = aj +ubj +vcj +uvdj for 0 ≤ j ≤ β −1. If C ⊆ Rα × Rβ is a Z [u]Z [u,v]-additive code, group isomorphic to Z2l0 × Zl1 × 1 2 2 2 2 2 Z4k0 × Z3k1 × Z2k2 × Zk3, then C is called a Z [u]Z [u,v]-additive code of type 2 2 2 2 2 2 (α,β;l ,l ;k ,k ,k ,k ) where l ,l ,k ,k ,k , and k are defined above. 0 1 0 1 2 3 0 1 0 1 2 3 5 3.1 Generator matrices of Z [u]Z [u,v]-additive codes 2 2 Theorem 3.1. Let C be a Z [u]Z [u,v]-additive code of type (α,β;l ,l ;k ,k ,k ,k ). 2 2 0 1 0 1 2 3 Then C is permutation equivalent to a Z [u]Z [u,v]-additive code with standard form 2 2 matrix. B T G = (3.2) S A " # where I B B B = l0 01 02 " 0 uIl1 uB12# 0 0 0 vT vT T = 01 02 "0 0 0 0 uvT12# 0 S S 01 02 0 0 uS S =  12 0 0 0   0 0 0      I A A A A k0 01 02 03 04 0 uI uA uA uA A =  k1 12 13 14  0 0 vI vA vA k2 23 24    0 0 0 uvI uvA   k3 34   . Here, B are matrices over R and T are matrices over R for 0 ≤ i < 2 and ij 1 ij 2 0 < j ≤ 2. Further, for 0 ≤ d < 2, 0 ≤ t < 4, 0 < q ≤ 4, S and A are matrices over dj tq R . Also, I and I are the identity matrices of size l and k , where 0 ≤ w ≤ 1 and 2 lw ky w y 0 ≤ y ≤ 3. Proof. First,byfocusingonthefirstαcoordinateswhichisaprojectionoftheZ [u]Z [u,v]- 2 2 additive code to a linear Z [u]-code and applying the necessary row operations together 2 with column operations(if neessary) we obtain the matrix B as in the upper left corner of the matrixEq.3.2. Next, by focusing on the last β coordinates and protecting the form of the matrix B, one consider linear codes over the ring Z [u,v] for which the generator 2 matrices are well known to be of the form A as in Eq.3.2. Finally, by protecting both the form of G as in Eq.3.2 which is referred to as the standard form matrix. Example 3.2. Let C be a Z [u]Z [u,v]-additive code of type (2,3;1,1;1,1,1,0) with 2 2 6 standard generator matrix of C is 1 1 0 0 0 0 v v 0 u u 0 0 0 0 uv   0 1 0 1 0 1 0 1   (3.3) 0 0 u 0 u u u u      0 0 0 0 0 v v v    0 0 0 0 0 0 0 uv     and C has |C| = 2221242322 = 212 = 4096 codewords. 3.2 Duality of Z [u]Z [u,v]-additive codes 2 2 An inner product for two elements x,y ∈ Rα ×Rβ is defined as 1 2 α α+β hx,yi = uv x y + x y ∈ R . i i j j 2 ! i=1 j=α+1 X X Let D be a Z [u]Z [u,v]- linear code of length n of type (α,β;l ,l ;k ,k ,k ,k ). We 2 2 0 1 0 1 2 3 denote by C the corresponding additive code, i.e., C = Φ−1(D). The additive dual code of C, denoted by C⊥, is defined in the standard way C⊥ = {y ∈ Rα ×Rβ|hx,yi = 0 for all x ∈ C} 1 2 The corresponding binary code Φ(C⊥) is denoted by C⊥ and is called the Z2[u]Z2[u,v]- dual code of C. The additive dual code C⊥ also an additive code, that is a subgroup of Rα ×Rβ. 1 2 Theorem 3.3. Let C be a Z [u]Z [u,v]-linear code of type (α,β;l ,l ;k ,k ,k ,k ). The 2 2 0 1 0 1 2 3 Z2[u]Z2[u,v]-dual code C⊥ is then of type (α,β;l0′,l1′;k0′,k1′,k2′,k3′), where ′ l = α−l −l 0 0 1 ′ l = l 1 1 ′ k = β −k 0 0 ′ k = β −k −k 1 1 2 ′ k = k 2 2 ′ k = β −k −k 3 2 3 3.3 Parity-check matrices of Z [u]Z [u,v]-additive codes 2 2 As in the classical case, the generator matrix of the dual code is important. In the following theorem, the standard form of the generator matrix of the dual code is presented. 7 Theorem 3.4. Let C be a Z [u]Z [u,v]-additive code of type (α,β;l ,l ;k ,k ,k ,k ) 2 2 0 1 0 1 2 3 with the standard form matrix defined in equation (3.2). Then the generator matrix for the additive dual code C⊥ is given by B¯ U H = (3.4) "V A¯+E# where B¯ = −B0t2 +B1t2B0t1 −B1t2 Iα−l0−l1 " −uB0t1 uIl1 0 # −Tt +Tt Bt +At Tt −Tt 0 02 12 01 34 01 12 −uTt 0 0 V =  01  0 0 0    0 0 0     −uSt At +u(Bt St −St ) −uSt 0 0 0 U = 12 01 12 01 02 12 " −uvSt 0 0 0 0# 01 Tt S 0 0 0 0 12 01 0 0 0 0 0 E =   0 0 0 0 0    0 0 0 0 0     l −um −uAt12 −uAt23 uIβ−k1−k2 0 A¯ = −vp −vAt vI 0 0  12 k2  −uvAt01 uvIβ−k2−k3 0 0 0    where l = −At +At (At +At At +At At +At At At )+At (At +At At )+At At 04 01 14 13 34 24 12 12 23 34 02 24 23 34 03 34 k = At +At At +At At +At At At 14 13 34 24 12 12 23 34 n = At +At At 24 23 34 m = At +At At +At At 03 02 23 01 12 p = At +At At 02 01 12 Proof. It is easy to check that HGt = 0. Besides, since we have |C||C⊥| = 22α+8β by checking thetypeofthematrixH weconcludethattherowsofH notonlyareorthogonal ⊥ to the rows of G but also generate the all code C . Hence, H is the desired matrix. 8 Example 3.5. Let C be a Z [u]Z [u,v]- additive code of type (2,3;1,1;1,1,1,0) with the 2 2 generator matrix given in Equation (3.3). Then, we can write the parity-check matrix of C as follows: 1 1 1 u u 0 0 0 u u 0 uv 0 0 0 0   1 1 0 0 0 0 1 1 H =  . u 0 0 u u u u 0     0 0 0 v v v 0 0   0 0 0 0 uv 0 0 0     It is clear that is of type (2,3;0,1;2,1,1,2) and has |C⊥| = 2128232222 = 216 = 65536 codewords. Z Z 4 The Structure of [u] [u,v]-additive cyclic code 2 2 In this section, we introduce the definition of a additive cyclic code and some algebraic structure. R [x] R [x] Let R [x] = 1 × 2 . α,β < xα −1 > < xβ −1 > A code C is cyclic if and only if its polynomial representation is an ideal. A additive code C is called a Z [u]Z [u,v]-additive cyclic code if any cyclic shift 2 2 of a codeword is also a codeword. i.e., (a0,a1,··· ,aα−1,b0,b1,··· ,bβ−1) ∈ C ⇒ (aα−1,a0,··· ,aα−2,bβ−1,b0,··· ,bβ−2) ∈ C. Theorem 4.1. If C is any Z [u]Z [u,v]-additive cyclic code, then C⊥ is also cyclic. 2 2 Proof. Let C be any Z [u]Z [u,v]-additive cyclic code. 2 2 ⊥ Suppose x = (a0,a1,··· ,aα−1,b0,b1,··· ,bβ−1) ∈ C , for any codeword y = (d0,d1,··· ,dα−1,e0,e1,··· ,eβ−1) ∈ C we have α−1 β−1 hx,yi = uv a d + b e = 0. i i j j (cid:18)i=0 (cid:19) j=0 Let S is a cycPlic shift, andPj = lcm(α,β). Then we have S(x) = (aα−1,a0,··· ,aα−2,bβ−1,b0,··· ,bβ−2) and Sj(y) = y for any y ∈ C. Since C be any Z [u]Z [u,v]-additive cyclic code, So we have 2 2 Sj−1(y) = (d1,d2,··· ,dα−1,d0,e1,e2,··· ,eβ−1,e0) ∈ C. 9 Hence 0 = hx,Sj−1(y)i = uv(a0d1 +a1d2 +···+aα−2dα−1 +aα−1d0) +(b0e1 +b1e2 +···bβ−2eβ−1 +bβ−1e0) = uv(aα−1d0 +a0d1 +a1d2 +···+aα−2dα−1) +(bβ−1e0 +b0e1 +b1e2 +···bβ−2eβ−1) = hS(x),yi ⊥ ⊥ Therefore, we have S(x) ∈ C , so C is a cyclic code. Let C be a Z [u]Z [u,v]-additive cyclic code, for any codeword 2 2 c = (a0,a1,··· ,aα−1,b0,b1,··· ,bβ−1) ∈ C can be representation with a polynomial, that is, c(x) = (a(x),b(x)) ∈ R [x] α,β Similarly, we introduce a new scalar multiplication. Now, we have the following scalar multiplication: for c (x) = (a (x),b (x)) ∈ R [x] and h(x) = p(x)+uq(x)+vr(x)+uvs(x) ∈ R [x], 1 1 1 α,β 2 define h(x)c (x) = ((p(x)+uq(x))a (x),h(x)b (x)). 1 1 1 Now, we define the homomorphism mapping: Ψ : R [x] → R [x] α,β 2 Ψ(c(x)) = Ψ(a(x),b(x)) = b(x). R [x] 2 It is clear that Image(Ψ) is an ideal in the ring and ker(Ψ) is also an < xβ −1 > R [x] ideal in 1 . And note that < xα −1 > Image(Ψ) = {b(x) ∈ R [x] : (a(x),b(x)) ∈ R [x]} 2 α,β R [x] ker(Ψ) = (a(x),0) ∈ R [x] : a(x) ∈ 1 . α,β < xα −1 > (cid:26) (cid:27) We have Image(Ψ) =< g(x)+up (x)+vp (x)+uvp (x),ua (x)+vq (x)+uvq (x),va (x)+ 1 2 3 1 1 2 2 uvr (x),uva (x) > 1 3 R [x] whereg(x),p (x),p (x),p (x),a (x),q (x),q (x),a (x),r (x),a (x) ∈ 2 and 1 2 3 1 1 2 2 1 3 < xβ −1 > a (x)|a (x)|a (x)|g(x)|xβ −1 mod 2, 3 2 1 xβ −1 a (x)|p (x) , 1 1 g(x) (cid:18) (cid:19) 10

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