LI Hui,HU Peng,LIU Xiu-sheng

(School of Mathematics and Physics,Hubei Polytechnic University,Huangshi 435003,China)

1 Introduction

Cyclic codes over finite rings are important class from a theoretical and practical viewpoint.It was shown that certain good nonlinear binary codes could be found as images of linear codes over Z4under the Gray map(see[1]).In[2],Zhu et al.studied constacyclic codes over ring F2+vF2,where v2=v.We in[3]generated ring F2+vF2to ring F2+uF2+vF2,where v2=v,u2=0,uv=vu=0,and studied the structure of cyclic of an arbitrary length n over this ring.

Boucher et al.in[4]initiated the study of skew cyclic codes over a noncommutative ring Fq[x,Θ],called skew polynomial ring,where Fqis a finite field and Θ is a field automorphism of Fq.Later,in[5],Abualrub and Seneviratne investigated skew cyclic codes over ring F2+vF2with v2=v.Moreover,Gao[6]and Gursoy et al.[7]presented skew cyclic codes over Fp+vFpand Fq+vFqwith different automorphisms,respectively.Recently,Yan,Shi and Sol`e in[8]investigated skew cyclic codes over Fq+uFa+vFq+vFq.

In this work,let R denote the ring Fq+uFq+vFqwhere u2=u,v2=v and uv=vu=0.In Section 2,we give some properties of ring R and de fine the Gray map ? from R to F3q.Moreover,we investigate some results about linear codes over R.In Section 3,we first give a sufficient and necessary condition which a code C is a skew cyclic code over R.We then characterize the generator polynomials of skew cyclic codes and their dual over R.Finally,in Section 4,we address the relationship of LCD codes between R and Fq.By means of the Gray map from R to,we obtain that Gray images of LCD codes over R are LCD codes over Fq.

2 Linear Codes Over R

The ring R is a finite commutative ring with characteristic p and it contains three maximal ideals which are

Let Rn={x=(x1,···,xn)|xj∈ R}be R-module.A R-submodule C of Rnis called a linear code of length n over R.We assume throughout that all codes are linear.

Let x,y∈Rn,the Euclidean inner product of x,y is de fined as follows

We call C⊥={x∈Rn|x·c=0,?c∈C}as the dual code of C.Notice that C⊥is linear if C is linear or not.

In[8],it was proved that for any linear code C over a finite Frobenius ring,

The Gray map ? :is de fined by ?(x)=(β(x1),···,β(xn))for x=(x1,···,xn),where β(a+ub+vc)=(a,a+b,a+c)for a+ub+vc ∈ R with a,b,c ∈ Fq.By using this map,we can de fine the Lee weight WLand Lee distance dLas follows.

De finition 2.1For any element x=(x1,···,xn)∈ Rn,we de fine WL(x)=WH(?(x)),where WHdenotes the ordinary Hamming weight for codes over Fq.The Lee distance dL(x,y)between two codewords x and y is the Lee weight of x?y.

Lemma 2.2The Gray map ? is a distance-preserving map from(Rn,Lee distance)to(F3n,Hamming distance)and also Fq-linear.

ProofFrom the de finition,it is clear that ?(x?y)= ?(x)? ?(y)for x and y ∈ Rn.Thus dL(x,y)=dH(?(x),?(y)).

For any x,y ∈ Rn,a,b ∈ Fq,from the de finition of the Gray map,we have ?(ax+by)=a?(x)+b?(y),which implies that ? is an Fq-linear map.

The following theorem is obvious.

Theorem 2.3If C is a linear code of length n over R,size qkand Lee distance dL,then ?(C)is a linear code over Fqwith parameters[3n,k,dL].

Theorem 2.4If C is a linear code of length n over R,then ?(C⊥)= ?(C)⊥.Moreover,if C is a self-dual code,so is ?(C).

ProofLet x1=a1+ub1+vc1,x2=a2+ub2+vc2∈C be two codewords,where a1,b1,c1,a2,b2,c2∈,and·be the Euclidean inner product on Rnor.Then and

It is easy to check that x1·x2=0 implies ?(x1)·?(x2)=0.Therefore

Combining(2.2)with(2.3),we get the desired equality.

Now,we mainly consider some familiar structural properties of a linear code C over R.The proof of following results can be found in[10],so we omit them here.

Let Ai(i=1,2,3)be codes over R.We denote

If C is a linear code of length n over R,we de fine that

It is easy to verity that Ci(i=1,2,3)are linear codes of length n over Fq.Furthermore,C=e1C1⊕e2C2⊕e3C3and|C|=|C1||C2||C3|.Throughout this paper,Ci(i=1,2,3)will be reserved symbols referring to these special subcodes.

According to above de finition and[10],we have the following theorem.

Theorem 2.5If C=e1C1⊕e2C2⊕e3C3is a linear code of length n over R,then

The next theorem gives a computation for minimum Lee distance dLof a linear code of length n over R.

Theorem 2.6If C=e1C1⊕e2C2⊕e3C3is a linear code of length n over R,then dL(C)=min{dH(C1),dH(C2),dH(C3)}.

ProofBy Theorem 2.3,we have dL(C)=dH(?(C)).

For any codeword x,it can be written as x=e1a+e2b+e3c,where a∈C1,b∈C2,c∈C3.Thus

This means that dL(C)=min{dH(C1),dH(C2),dH(C3)}.

3 Skew Cyclic Codes Over R

Let R=Fq+uFq+vFq,where q=pm,p is a prime.For integer 0≤s≤m,we consider the automorphisms

In this section,we first de fine skew polynomial rings R[x,Θs]and skew cyclic codes over R.Next,we investigate skew cyclic codes over R through a decomposition theorem.

De finition 3.1We de fine the skew polynomial ring as R[x,Θs]={a0+a1x+ ···+,where the coefficients are written on the left of the variable x.The addition is the usual polynomial addition and the multiplication is de fined by the rule xa=Θs(a)x(a∈R).

It is easy to prove that the ring R[x,Θs]is not commutative unless Θsis the identity automorphism on R.

De finition 3.2A linear code C of length n over R is called skew cyclic code if for any codeword c=(c0,c1,···,cn?1) ∈ C,the vector Θs(c)=(Θs(cn?1),Θs(c0),···,Θs(cn?2))is also a codeword in C.

The following theorem characterizes skew cyclic codes of length n over R.

Theorem 3.3Let C=e1C1⊕e2C2⊕e3C3be a linear code of length n over R.Then C is a skew cyclic code of length n over R if and only if C1,C2and C3are skew cyclic codes of length n over Fq,respectively.

ProofSuppose that xi=e1ai+e2bi+e3ci,where ai,bi,ci∈ Fq,i=0,1,···,n ? 1,and x=(x0,x1,···,xn?1).Then

Set a=(a0,a1,···,an?1),b=(b0,b1,···,bn?1),c=(c0,c1,···,cn?1),thus x=e1a+e2b+e3c and a∈C1,b∈C2,c∈C3.If C is a skew cyclic code of length n over R,then

Therefore Θs(a)∈ C1,Θs(b)∈ C2,Θs(c)∈ C3.This means that C1,C2and C3are skew cyclic codes.

Conversely,if Ciare skew cyclic codes over Fq,then

This implies that C is a skew cyclic code over R.

Theorem 3.4Let C=e1C1⊕e2C2⊕e3C3be a skew cyclic code of length n over R.Then

(2)There is a unique polynomial g(x)such that C= 〈g(x)〉and g(x)|xn? 1,where g(x)=e1g1(x)+e2g2(x)+e3g3(x).Moreover,every left submodule of R[x,Θs]/〈xn? 1〉is principally generated.

Proof(1)Since Ci= 〈gi(x)〉? Fq[x,Θs]/〈xn?1〉for i=1,2,3 and C=e1C1⊕e2C2⊕e3C3,C= 〈c(x)|c(x)=e1f1(x)+e2f2(x)+e3f3(x),fi(x)∈ Ci,i=1,2,3〉.Thus

On the other hand,for any e1r1(x)g1(x)+e2r2(x)g2(x)+e3r3(x)g3(x)∈ 〈e1g1(x),e2g2(x),e3g3(x)〉? R[x,Θs]/〈xn? 1〉,where r1(x),r2(x)and r3(x)∈ R[x,Θs]/〈xn? 1〉,there exist

s1(x),s2(x)and s3(x)∈ Fq[x,Θs]/〈xn? 1〉such that e1r1(x)=e1s1(x),e2r2(x)=e2s2(x)and e3r3(x)=e3s3(x).Hence

which implies that〈e1g1(x),e2g2(x),e3g3(x)〉? C.Therefore C= 〈e1g1(x),e2g2(x),e3g3(x)〉.

(2)Obviously,〈e1g1(x)+e2g2(x)+e3g3(x)〉? 〈e1g1(x),e2g2(x),e3g3(x)〉.

Note that e1g(x)=e1g1(x),e2g(x)=e2g2(x)and e3g(x)=e3g3(x),we have C ? 〈g(x)〉.Therefore,C= 〈g(x)〉.

Since g1(x),g2(x)and g3(x)are monic right divisors of xn?1,there exist h1(x),h2(x)and h3(x)∈ Fq[x,Θs]/〈xn?1〉such that xn?1=h1(x)g1(x)=h2(x)g2(x)=h3(x)g3(x).Therefore xn?1=[e1h1(x)+e2h2(x)+e3h3(x)]g(x).It follows that g(x)|xn?1.The uniqueness of g(x)can be followed from that of g1(x),g2(x)and g3(x).

Let g(x)=g0+g1x+···+gkxkand h(x)=be polynomials in Fq[x,Θs]such that xn?1=h(x)g(x)and C be the skew cyclic code generated by g(x)in Fq[x,Θs].Then the dual code of C is a skew cyclic code generated by the polynomial

Corollary 3.5Let C1,C2,C3be skew cyclic codes of length n over Fqand g1(x),g2(x),g3(x)be their generator polynomials such that

in Fq[x,Θs].If C=e1C1⊕e2C2⊕e3C3,then

(1)is also a skew cyclic code of length n over R,where

(2)C is a self-dual skew cyclic code over R if and only if C1,C2and C3are self-dual skew cyclic codes of length n over Fq.

Proof(1)In light of Theorem 2.5,we obtain

(2)C is a self-dual skew cyclic code over R if and only if g(x)=(x),i.e.,g1(x)=(x),g2(x)=(x)and g3(x)=(x).Thus C is a self-dual skew cyclic code over R if and only if C1,C2and C3are self-dual skew cyclic codes of length n over Fq.

Example 1Let ω a primitive element of F9(where ω =2ω +1)and Θ be the Frobenius automorphism over F9,i.e.,Θ(a)=a3for any a∈F9.Then

Let g1(x)=2+(2+ ω)x+(1+2ω)x3+x4and g2(x)=g3(x)=2+x+(2+2ω)x2+x3.Then C1= 〈g1(x)〉and C2=C3= 〈g2(x)〉are skew cyclic codes of length 6 over F9with dimensions k1=2,k2=k3=3,respectively.Take g(x)=e1g1(x)+e2g2(x)+e3g3(x),then C is a skew cyclic code of length 6 over R.Thus the Gray image of C is a[18,8,4]code over F9.

4 LCD Codes over R

Linear complementary dual codes(which is abbreviated to LCD codes)are linear codes that meet their dual trivially.These codes were introduced by Massey in[12]and showed that asymptotically good LCD codes exist,and provide an optimum linear coding solution for the two-user binary adder channel.In[13],Sendrier indicated that linear codes with complementary-duals meet the asymptotic Gilbert-Varshamov bound.They are also used in counter measure to passive and active side channel analyses on embedded cryto-systems(see[14]).In recent,we in[15]investigated LCD codes finite chain ring.Motivated by these works,we will consider the LCD codes over R.

Suppose that f(x)is a monic(i.e.,leading coefficient 1)polynomial of degree k with f(0)=c0.Then by monic reciprocal polynomial of f(x)we mean the polynomial(x)=c?1f?(x).

We recall a result about LCD codes which can be found in[16].

Proposition 4.1If g1(x)is the generator polynomial of a cyclic code C of length n over Fq,then C is an LCD code if and only if g1(x)is self-reciprocal(i.e.,(x)=g1(x))and all the monic irreducible factors of g1(x)have the same multiplicity in g1(x)and in xn?1.

Theorem 4.2If C=e1C1⊕e2C2⊕e3C3is a linear code over R,then C is a LCD code over R if and only if C1,C2and C3are LCD codes over Fq.

ProofC is a LCD code over R if and only if C∩C⊥={0}.By Theorem 2.5,we know that C∩C⊥={0}if and only if C1∩C⊥1={0},C2∩C⊥2={0},and C3∩C⊥3={0},i.e.,C1,C2and C3are LCD codes over Fq.

By means of Proposition 4.1 and above theorem,we have the following corollary.

Corollary 4.3Let C=e1C1⊕e2C2⊕e3C3is a cyclic code of length n over R,and let C1= 〈g1(x)〉,C2= 〈g2(x)〉and C3= 〈g3(x)〉be cyclic codes of length n over Fq.Then C is a LCD code over R if and only if gi(x)is self-reciprocal(i.e.,(x)=gi(x))and all the monic irreducible factors of gi(x)have the same multiplicity in gi(x)and in xn?1 for i=1,2,3.

ProofIf x ∈ C ∩ C⊥,then x ∈ C and x ∈ C⊥.It follows that ?(x)∈ ?(C)and ?(x)∈ ?(C⊥).Hence ?(C ∩C⊥)? ?(C)∩?(C⊥).

On the other hand,if ?(x)∈ ?(C)∩ ?(C⊥),then there are y ∈ C and z∈ C⊥such that ?(x)= ?(y)= ?(z).Since ? is an injection,x=y=z∈ C ∩C⊥,which implies that

Thus ?(C)∩ ?(C⊥)= ?(C ∩C⊥).

By Theorem 2.3,we ?(C ∩C⊥)= ?(C)∩?(C)⊥.It follows that C ? Rnis LCD if and only if the linear code ?(C)is LCD.

Example 2x4?1=(x+1)(x+2)(x+w2)(x+w6)in F9.Let g1(x)=g2(x)=g2(x)=x+1.Then C1=C2=C3= 〈g1(x)〉are LCD cyclic codes over F9with parameters[4,3,2],respectively.Suppose that C=e1C1⊕e2C2⊕e3C3is a cyclic code of length n over R.By Theorem 2.6 and Theorem 4.5,?(C)is a LCD code with parameters[12,9,2],which is an optimal code.

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