Derong XIE Qunying LIAO

1College of Mathematical Science,Sichuan Normal University,Chengdu 610066,China.E-mail: qunyingliao@sicnu.edu.cn derongxie@yahoo.com

1 Introduction

Let q be a power of the prime p and Fqbe the finit field of q elements.In 1957,the concept of cyclic codes over Fqwas proposed (cf.[14]).Cyclic codes over finite fields,as a class of good linear codes,have attracted extensive attentions due to their special algebraic properties,decoding algorithms,and easy realization etc.In 1967,the concept of negacyclic codes over Fqwas given (cf.[1]).Later,scholars generalized cyclic codes over finite fields to be constacyclic codes over finite fields.Fixed u ∈F∗q,the linear code C with length n over Fqis a u-constacyclic code if for any c=(c0,c1,··· ,cn−1)∈C,the u-constacyclic shift(ucn−1,c0,c1,··· ,cn−2)∈C.In particular,when u=1,C is a cyclic code; when u=−1,C is a negacyclic code.It’s well-known that self-orthogonal and self-dual cycle codes over finite fields are both useful in cryptography and coding theory due to their many good algebraic properties.In 2011,some sufficient and necessary conditions for the existence of self-dual cyclic codes over Fqwere obtained (cf.[10]).In 2014,a sufficient and necessary condition for the existence of nontrivial self-orthogonal cyclic codes over Fqwas obtained and then the corresponding explicit enumerating formula were determined (cf.[11]).

On the other hand,in recent years,codes over finite rings are also interesting since the binary image of a linear code over Z4is a binary code (not necessarily linear)(cf.[4–7,12]).

Definition 1.1(cf.[5])Let R be a finite ring,a code C with length n over R is a nonempty subset of Rn,and the ring R is referred to as the alphabet of the code.If this subset is,in addition,an R-submodule of Rn,then C is called linear.For a unit λ of R,the λ-constacyclic(λ-twisted)shift τλon Rnis the shift and a linear code C is said to be λ-constacyclic if τλ(C)=C.It means that C is closed under the λ-constacyclic shift τλ.In case λ=1,those λ-constacyclic codes are called cyclic codes,and when λ=−1,such λ-constacyclic codes are called negacyclic codes.

Proposition 1.1(cf.[5])Let R be a finite ring,a linear code C with length n is λconstacyclic over R if and only if C is an ideal of the ring

Definition 1.2(cf.[5,8])Given two n-tuples x=(x0,x1,··· ,xn−1),y=(y0,y1,··· ,yn−1)∈Rn,their inner product or dot product is defined as usual:

evaluated in R.Two n-tuples x,y are called orthogonal if x·y=0.For a linear code C over R,its dual code C⊥is the set of n-tuples over R which are orthogonal to all codewords of C,i.e.,

A code C is self-orthogonal if C ⊆C⊥,and it is self-dual if C=C⊥.

Definition 1.3(cf.[13])Let x ∈R,[x]denotes the largest integer less than x,the function[x]is called the Gauss function.

Proposition 1.2(cf.[4])Suppose that R is a finite ring,λ is a unit of R and C is an λ-constacyclic code over the finite ring R with identity,then C⊥is an λ−1-constacyclic code over R.

In 2003,Blackford [2]studied negative cyclic codes with even length over Z4.In 2009,the structure of negative cyclic codes with even length and their dual codes over the finite chain ring Z2awas obtained,where a is a positive integer (cf.[15]).In 2013,a (1+ωγ)-constacyclic code of arbitrary length over the general finite chain ring were constructed (cf.[3]).

The present paper continues to the study,and discusses constacyclic codes over the finite non-chain ring Zp1p2···pt,where p1,p2,··· ,ptare distinct primes.A sufficient and necessary condition for the existence of both constacyclic codes and non-trivial self-orthogonal cyclic codes over Zp1p2···ptare obtained,and then the explicit enumerating formula for the numbers of these codes is given.In fact,the following main results are proved.

Theorem 1.1Let p1,p2,··· ,ptbe distinct primes,λ be a unit of Zp1p2···pt.Suppose that C is a code with length n over Zp1p2···pt,then

(1)C is an λ-constacyclic code if and only if there exist some λi-constacyclic codes Ciwith length n over Zpisuch thatwhere λi≡λ(mod pi)(1 ≤i ≤t);

(2)C is a cyclic code if and only if for any i=1,··· ,t,Ciis a cyclic code over Zpi;

(3)C is a self-orthogonal (self-dual)cyclic code if and only if for any i=1,··· ,t,Ciis a self-orthogonal (self-dual)cyclic code over Zpi.

Without loss of generality,if C={0},then a code C over Zp1p2···ptis trivial.Otherwise,C is non-trivial.The following Theorem 1.2 gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over Zp1p2···pt.

Theorem 1.2There exists a non-trivial self-orthogonal cyclic code with length n over Zp1p2···ptif and only if there is at least one pi,such that one of the following conditions is satisfied.

(1)gcd(n,pi)1.

(2)If gcd(n,pi)=1,then 2 ∤ordn(pi).

(3)If gcd(n,pi)=1 and 2|ordn(pi),then

It is well-known that,for a polynomial

the reverse polynomial is

Especially,there exists some α ∈F∗qsuch that f(x)=αg∗(x),then f(x)and g(x)are a pair reciprocal polynomials.Furthermore,if f(x)=αf∗(x),then f(x)is an introspect polynomial.On the other hand,for any positive integer n with gcd(n,q)=1,xn−1 has the unique irreducible factorizations over Fqas follows:where fi(x)(1 ≤i ≤k)is irreducible introspect and hj(x)(1 ≤j ≤l)is irreducible over Fq(cf.[8,11]).Based on this,the explicit enumerating formula for the number of non-trivial self-orthogonal cyclic codes over the ring Zp1p2···ptis obtained.And then there is no self-dual cyclic code with any length over Zp1p2···ptwhen t ≥2.

Theorem 1.3Let t ≥2 be an integer,p1,p2,··· ,ptbe distinct primes.Suppose thatand xni−1 has the unique irreducible factorizations over Fpias follows:

Set

then

and the number of non-trivial self-orthogonal cyclic codes with length n over Zp1p2···ptis

where [·]is the Gauss function.

Theorem 1.4Let t ≥2 be an integer,p1,p2,··· ,ptbe distinct primes.Then there is no self-dual cyclic code with any length over Zp1p2···pt.

2 Preliminaries

Some preliminaries are needed before proving our main results.Let q be a power of the prime p,n be a positive integer,and ordn(q)be the order of q modulo n.Without loss of generality,set ord1(q)=1.For convenience,all rings in this paper have identities.

Definition 2.1(cf.[9])Let R1,R2,··· ,Rtbe rings and

For (a1,a2,··· ,at),(b1,b2,··· ,bt)∈R,define two operations “+” and “∗” over R as follows:

easily to see that (R,+,∗)is a ring and called to be the direct sum of Ri(i=1,2,··· ,t).

Proposition 2.1(cf.[9])

(1)Let Z be the integer ring and m ∈N+,then

(2)Let θ : R →T be a ring homomorphism,then θ is a monomorphism if and only if ker(θ)=0,where ker(θ)={a ∈R|θ(a)=0}.

(3)Let R be a commutative ring,and Ii(i=1,2,··· ,t)be pairwise coprime ideals of R.Ifthen there is a ring isomorphism

Proposition 2.2(cf.[11])Let n ∈Z+and q be a power of the prime p.

(2)If gcd(n,q)=1,then there exists a non-trivial self-orthogonal cyclic code with length n if and only if n>2,there exists a positive divisor d ≥3 of n and one of the following is true:

(1◦)2 ∤ordd(q);

(2◦)if 2|ordd(q),then

Proposition 2.3(cf.[11])Let n ∈Z+,q be a power of the prime p with gcd(n,p)=1.Then there exists a non-trivial self-orthogonal cyclic code with length n over Fqif and only if n>2 and one of the following is true:

(1)2 ∤ordn(q);

(2)if 2|ordn(q),then n

Proposition 2.4(cf.[11])Let n ∈Z+,q be a power of the prime p with gcd(n,p)=1,and ϕ(n)be the Euler function of n.If xn−1 ∈Fq[x]has the factorizations as (∗),and

then

and the number of irreducible factors for xn−1 over Fqis

Proposition 2.5(cf.[8])Let n ∈Z+,q be a power of the prime p,and xn−1=g(x)h(x)with g(x),h(x)∈Fq[x].Then the cyclic code C=(g(x))with length n over Fqis self-orthogonal if and only if h∗(x)|g(x),i.e.,g(x)=m(x)h∗(x),xn−1=m(x)h(x)h∗(x),where h∗(x)is the reverse polynomial of h(x).

Proposition 2.6(cf.[11])Let p be a prime and n=prn0with gcd(n0,p)=1.Suppose q is a power of the prime p,and xn0−1 ∈Fq[x]has the unique irreducible factorizations over Fqas follows:

Then the number of non-trivial self-orthogonal cyclic codes with length n over Fqis

Proposition 2.7(cf.[11])Let q be a power of the prime p.Suppose that there exists a self-dual cyclic code C with length n over Fq,then 2|gcd(n,q).

The following two lemmas are important to prove our main results.

Lemma 2.1Let t be a positive integer,R,R1,··· ,Rtbe commutative rings with identities.If there is a ring isomorphism between R andthen there is a polynomial ring isomorphism

ProofLet ϕ :be a ring isomorphism with ϕ(aj)=(a1j,a2j,··· ,atj),where aj∈R,aij∈Ri(i=1,2,··· ,t).Define the map ϕ′:

i.e.,ϕ′it’s easy to show that ϕ′is bijective.Since ϕ is a ring isomorphism,thus for any we have

and

this means that ϕ′is a ring homomorphism.

Thus we complete the proof of Lemma 2.1.

Lemma 2.2Let R,R1,··· ,Rtbe commutative rings,and φ be a ring isomorphism between R andIf I is an ideal of R,i.e.,IR,and J=φ(I),then we have

Proof(1)Since φ is a ring isomorphism between R andand I ⊳R,thus J=φ(I)⊳Furthermore,for (r1,r2,··· ,rt)∈J,we have

namely,ri∈Ii(i=1,2,··· ,t),and then (r1,r2,··· ,rt)On the other hand,forwe can get

Note that for any i=1,2,··· ,t,from 0 ∈Iiwe know that Iiis not empty.Now for a,b ∈Ii,from the definition of Ii,we have

which means that a −b ∈Ii.Secondly,for any ri∈Ri(i=1,2,··· ,t),we know that

and

i.e.,ari,ria ∈Ii(1 ≤i ≤t).Hence Ii⊳Ri(i=1,2,··· ,t).Thus we complete the proof of(1).

(2)For any r ∈R,denote φ(r)=(r1,r2,··· ,rt),where ri∈Ri(1 ≤i ≤t).Now define a map φ′:

Note that if φ′(r+I)=(0,0,··· ,0),i.e.,

equivalently,ri∈Ii(i=1,2,··· ,t),then

which means that r ∈I,i.e.,r+I=0,thus from (2)of Proposition 2.1,φ′is injective.

Now for any (r1+I1,r2+I2,··· ,rt+It)there exists some r ∈R such that φ(r)=(r1,r2,··· ,rt)since φ is an epimorphism,hence we can get

i.e.,φ′is an epimorphism.

Furthermore,for s ∈R,set φ(s)=(s1,s2,··· ,st)and φ(rs)=((rs)1,(rs)2,··· ,(rs)t).Since φ is a ring isomorphism from R towe have

Now for any r+I,s+I ∈R/I,we have

and

From (3.1)and (3.3),we know that

This means that φ′is a ring homomorphism.

From the above,φ′is a ring isomorphism from R/I toThis completes the proof of (2).

3 The Proofs of Our Main Results

In this section,we give the proofs of the main results.

Proof of Theorem 1.1(1)Since p1,p2,··· ,ptare distinct primes,(i=1,2,··· ,t)are pairwise coprime ideals of Z.From (1)and (3)of Proposition 2.1 and Lemma 2.1,we can get the following two ring isomorphisms:

and

Thus by Lemma 2.2 we have

(2)By taking λ=1 in (1),we can get (2).

(3)By Proposition 1.1,there exists a ring isomorphism

Now,for any ci∈Ciand di∈Di(i=1,2,··· ,t),we have

Note that τ is an isomorphism and so τ−1is also an isomorphism.Thus τ−1(c)∈C and τ−1(d)∈C⊥,namely,τ−1(c)τ−1(d)=0.While τ−1is an isomorphism,hence

i.e.,cidi=0 (1 ≤i ≤t),this means that Di⊆C⊥i(i=1,2,··· ,t).

On the other hand,for any c′i∈C⊥i(i=1,2,··· ,t),i.e.,cic′i=0.By (1)–(2)of Theorem 1.1,there exist some ci∈Ci(i=1,2,··· ,t)and c′∈τ−1(C⊥1,C⊥2,··· ,C⊥t)such that

Thus

Hence from cic′i=0 (1 ≤i ≤t)and τ−1is an isomorphism,we have

i.e.,c′∈C⊥.Now from τ(C⊥)=(D1,D2,··· ,Dt),we know that

i.e.,c′i∈Di(i=1,2,··· ,t),thus C⊥i⊆Di(i=1,2,··· ,t).

Therefore C⊥i=Di(i=1,2,··· ,t),i.e.,

Thus from τ is an isomorphism and (3.4),we can obtain:

C is a self-orthogonal cyclic code ⇔C ⊆C⊥⇔τ(C)⊆τ(C⊥)

In particular,C is self-dual,i.e.,C=C⊥if and only if for any i=1,2,··· ,t,Ci=C⊥i,i.e.,Ciis self-dual.

This completes the proof of (3).

Proof of Theorem 1.2By Theorem 1.1 we know that C is a self-orthogonal cyclic code over Zp1p2···ptif and only if there exist self-orthogonal cyclic codes Ciover Zpi(i=1,2,··· ,t)such that Note that,C={0} if and only if Ci={0} (i=1,2,··· ,t).Now by Theorem1.1,C is non-trivial self-orthogonal if and only if there exist some Ci⊆(i=1,2,··· ,t)such that Ciis a non-trivial self-orthogonal cyclic code.By Propositions 2.2–2.3,this is equivalent to that there exist some pi(i=1,2,··· ,t)such that one of the following conditions is true.

(1)gcd(n,pi)1;

(2)If gcd(n,pi)=1,then 2 ∤ordn(pi);

(3)If gcd(n,pi)=1 and 2|ordn(pi),then

This completes the proof of Theorem 1.2.

Proof of Theorem 1.3Note that for any i=1,2,··· ,t,Zpiis a finite field since pi(1 ≤i ≤t)is a prime.Thus by Proposition 2.6,the number of non-trivial self-orthogonal cyclic codes with length n over Zpiis

Now from (3)of Lemma 2.1,the number of self-orthogonal cyclic codes with length n over Zp1p2···ptis

Note that C={0} if and only if Ci={0} (i=1,2,··· ,t),thus from Theorem 1.2 and Proposition 2.6,we immediately have Theorem 1.3.

Proof of Theorem 1.4From(3)of Theorem 1.1,there exists a self-dual cyclic code with length n over Zp1p2···ptif and only if for any i=1,2,··· ,t,there exists a self-dual cyclic code with length n over Zpi.Note that t ≥2 is an integer and p1,p2,··· ,ptare distinct primes,hence there is no self-dual cyclic code over Zp1p2···ptby Proposition 2.7.

4 Examples

In this section,by using elementary methods and techniques,one can get the number of non-trivial self-orthogonal cyclic codes over Zp1p2···ptbasing on Theorem 1.3.

Example 4.1For n=10=2×5,we have gcd(10,3)=1.Then by Theorem 1.1,we know that there exists a self-orthogonal cyclic code C over Z6if and only if there exists a self-orthogonal cyclic code C1over Z2and a self-orthogonal cyclic code C2over Z3,such that CC1⊕C2.

Note that ord5(2)=4,ord2(3)=1,ord5(3)=4 and ord10(3)=4,then by Theorem 1.3,we have k1=2,k2=4,l1=0 and l2=0.Thus the number of non-trivial self-orthogonal cyclic codes over Z6is

Furthermore,by Theorem 1.4,there doesn’t exist any self-dual cyclic code over Z6.

On the other hand,the canonical decomposition of x10−1 over Z2is

And the canonical decomposition of x10−1 over Z3is

Now set

and

which are not self-dual cyclic.

Example 4.2For n=9=32,we have gcd(9,5)=1.Now by Theorem 1.1,we know that there exists a self-orthogonal cyclic code C over Z15if and only if there exists a self-orthogonal cyclic code C1over Z3and a self-orthogonal cyclic code C2over Z5,such that CC1⊕C2.

Note that ord3(5)=2 and ord9(5)=6,then by Theorem 1.3,we have

Thus the number of non-trivial self-orthogonal cyclic codes over Z15is

Furthermore,by Theorem 1.4,there doesn’t exist any self-dual cyclic code over Z15.

On the other hand,the canonical decomposition of x9−1 over Z5is

Now set

Then by Proposition 2.5,C11=are non-trivial self-orthogonal cyclic codes over Z3and there doesn’t exist any non-trivial self-orthogonal cyclic code over Z5.Thus by Theorem 1.1,there are eight non-trivial self-orthogonal cyclic codes over Z15as follows:

which are not self-dual cyclic.

5 Conclusion

It’s well-known that the polynomial ring Zp1p2···pt[x]is not a unique factorization domain.Hence to study constacyclic codes over R=Zp1p2···ptis difficult basing on polynomial factorizations.By constructing an isomorphic between R and Zpi(i=1,2,··· ,t),to study constacyclic codes over R is reduced to study the corresponding constacyclic codes over finite fields Zpi,which is much easier.Based on this,the present paper studies constacyclic codes over Zp1p2···ptand obtains some good results.