The Self-dual Codes over Formal Power Series Rings

LIU Xiu-sheng,LIU Hua-lu

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

The Self-dual Codes over Formal Power Series Rings

LIU Xiu-sheng,LIU Hua-lu

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

The codes of formal power series ringsand finite chain rings Ri={a0+a1γ+···+ai?1γi?1|ai∈F}have close relationship in lifts and projection.In this paper,we study self-dual codes over R∞by means of self-dual codes over Ri,and give some characterizations of self-dual codes over R∞.

self-dual codes;γ-adic codes;projective codes

§1. Introduction

Self-dualcodes are an important class ofcodes from many perspectives.They have interesting connections to design theory,combinatorics,lattice theory,invariant theory,modular forms and number theory.

See[1],[8-9]for descriptions of these connections.In particulary,self-dual codes over rings have been proven to be important,especially in connection with lattices,modular forms and number theory.

Originally Self-dual codes were studied over the binary fields F2.More recently,mathematicians have become interested in codes over various finite rings by the paper written by Hammons et al(see[6]).In this paper,it is proved that some interesting non-linear codes,such as the Kerdock,Preparata and Goethal codes can be viewed as linear codes over Z4via theGray map fromto.Following this,a large series of papers studying codes over rings appeared.

In[4],Calderbank and Sloane studied codes over the p-adics and also gave a description on the lifts of codes over Zpto Zpeand to the p-adics.Later,in[7],Dougherty,Kim and Park studied these codes further and also found the weight enumerator of this class of codes.An important class of rings over which codes have been studied is the class ofchain rings,especially Zpe,which are a special case of chain rings.

In[3],Dougherty,Liu and Park define a series of chain rings and introduce the concept of γ-adic codes.They then also study the lifts and projection over this class of rings.In this paper,we focus on self-dual codes over this class of rings.We study self-dual codes ofγ-adic codes via the projection ofγ-adic codes.

Throughout this paper we let R be a finite commutative ring with identity 1/=0.Let Rn={x=(x1,···,xn)|xj∈R}be an R-module.An R-submodule C of Rnis called a linear code of length n over R.

Let x,y∈Rn.The inner product of x,y is defined as the following

We call

as the orthogonalcode of C.Notice that C⊥is linear if C is linear or not.

If C?C⊥,then C is called self-orthogonal.Moreover,if C=C⊥,then C is called self-dual.

§2. Preliminaries

We begin with the definition and some properties of finite chain rings.

Defi nition 1 A finite commutative ring with identity 1/=0 is called a finite chain ring if its ideals are linearly ordered by inclusion.

A simple example of a finite chain ring is the ring Zpaof integers modulo pa,for some prime p and a≥1.A finite chain ring is,obviously,a localring.It is well-known,and not diffi cult to prove,that a ring is a finite chain ring if and only if it is a finite localprincipalidealring.Let γbe a fixed generator of the maximal ideal of a finite chain ring R.Thenγis nilpotent and let e be its nilpotency index i.e.,the smallest positive integer such thatγe=0.

Let|R|denote the cardinality of R and R×the multiplicative group of all units in R.Let F=R/γR=R/〈γ〉be the residue field with characteristic p,where p is a prime number.We know that|F|=q=prfor some integers q and r and|F×|=pr?1.

Let R be a finite chain ring.We know from[2]that the generator matrix for a code C overR is permutation equivalent to a matrix of the following form

The matrix G above is called the standard generator matrix form of the code C.It is immediate that a code C with this generator matrix has cardinality

In this case,the code C is said to have type

For an arbitrary positive integer i,we define Rias

whereγi?1/=0,butγi=0 in Riand define two operations over Ri

It is easy to prove that all the Riare chain rings with maximalideal〈γ〉(see[3]).

Further,we define R∞as the ring of formal power series as follows

We know that for any two elements,their sum and product are the following

Then the ring R∞is a principalideal domain.

Let C be a nonzero linear code over R∞of length n.We know from[3]that any generator matrix of C is permutation equivalent to a matrix of the following form

where 0≤m0＜ m1＜ ···＜ mr?1for some integer r.The column blocks have sizes k0,k1,···,krand the kiare nonnegative integers adding to n.

Remark 1 A code C with generator matrix of the form given in Equation(2.8)is said to be of type

where k=k0+k1+···+kr?1is called its rank and kr=n?k.

A code C of length n with rank k over R∞is called aγ-adic[n,k]code.

The following two definitions will be used throughout the paper.

Defi nition 2 For any positive integer i＜∞,we define a map as follows

Let a,b be two arbitrary elements in R∞.It is easy to get that

In the case,the mapΨiis said to a canonical projection from R∞to Ri.

We note that the mapΨican be extended naturally fromto

Defi nition 3 If C is a[n,k]γ-adic code,then for any i＜∞,we callΨi(C)a projection code of C over Ri.We denoteΨi(C)by Ci.

§3. Self-dualγ-adic Code

In this section,we describe self-dual over R∞.We fix the ring R∞with

and R1=Fq,where q=prfor some prime p and nonnegative integer r.The following Lemma can be found from[3].

Lemma 1 If C is a linear code over R∞,then C⊥has type 1mfor some m.

Lemma 2 If C is a self-dual code of length n over R∞,then C has type 1kfor some k and n=2k.

Proof Sine C is self-dual,we have that C=C⊥.By Lemma 1,the code C⊥has type 1kfor some k.Hence we have that k=n?k,this gives that n=2k.

Remark 2 Let C be a code over R∞.We know that C?(C⊥)⊥.But in general C/=(C⊥)⊥.For example,let C be a code of length 2 over R∞generated by(γi,γi),where 1≤i＜∞.We know C⊥={(a,?a)|a∈R∞}and then(C⊥)⊥=〈(1,1)〉.Since(1,1)/∈C, This implies that

The following Lemma can be found in[3].

Lemma 3 Let C be a linear code of length n over R∞.Then C=(C⊥)⊥if and only if C has type 1kfor some integer k.

With the lemmas above,we shall prove the following theorems.

Theorem 1 If C is a self-dual code of length n over R∞,thenΨi(C)is a self-dual code of length n over Rifor all i＜∞.

Proof First we show thatΨi(C)is a self-orthogonal code of length n over Rifor all i＜∞.

For any v=(v1,···,vn),w=(w1,···,wn)∈C,we have[v,w]=0 since C=C⊥.This implies that

HenceΨi(C)is a self-orthogonalcode of length n over Rifor all i＜∞.

It remains to show thatΨi(C)=Ψi(C)⊥.To show this equation,it suffi ces to show that Ψi(C⊥)=Ψi(C)⊥since C=C⊥.

In fact,for any x∈Ψi(C⊥)and y∈Ψi(C),we know that there exist x′∈C⊥and y′∈C such that x=Ψi(x′)and y=Ψi(y′).This gives that

This means thatΨi(C⊥)?Ψi(C)⊥.

By Lemma 2,C and C⊥have typeThusΨi(C⊥)has typeand(Ψi(C))⊥has typeTherefore|Ψi(C⊥)|=|(Ψi(C))⊥|.This implies thatΨi(C)=Ψi(C)⊥.

Theorem 2 Let C be a nonzero linear code of length n over R∞.Assume thatΨj(C)is a self-orthogonal code of length n over Rjfor all j＜∞,and there existsΨi(C)is a self-dual code of length n over Rifor some i,then C is a self-dual code over R∞.

Proof Firstwe show that C is a self-orthogonalcode oflength n over R∞by contradiction.

Suppose C is not a self-orthogonal code over R∞.Then there existsα∈C such thatα/∈C⊥.Therefore,we can find thatβ∈C such that

Without loss of generality,let ak?1be the first nonzero element among a0,a1,···,ak?1, ak,···,we have

On one hand,

on the other hand

This implies that akγk?1=0.i.e.ak?1=0,This is a contradiction.

In the following,we show C=C⊥.

Let C be a nonzero linear code over R∞,we get that C⊥has type 1mfor some integer m by Lemma 1.According to the proof above,we have C?C⊥.In this case,assume that C has type 1s,thenΨi(C)has type 1sandΨi(C⊥)so does.

SinceΨi(C⊥)=Ψi(C)⊥,Ψi(C)=Ψi(C)⊥.ThenΨi(C)=Ψi(C⊥)and s=m.Therefore C=C⊥.Hence C is a self-dualcode over R∞.

§4. Self-dual Code over Principal Ideal Rings

In this section,we study codes over principalidealrings by the generalized Chinese Remainder Theorem(see[3,5]).

We know that A is a principal ideal ring.Let

It is easy to see that all Ijare idealof A andWe have that I1+···+Is=A and for each Ik,where 1≤k≤s,we have that

Hence by Theorem 2.24 in[10],there is an isomorphism as follows

We denote the isomorphism above as

We denote the inverse isomorphism ofΦby CRT,that is

We denote the image of CRT as CRT

Then C is called the Chinese product of codes

This gives that all the ringsare principal ideal rings.In particular,We denote CRTby

Let i be a positive integer such that i＜∞.By the mapΨiin Equation(2.9),we get the following map which is also denoted byΨi,

This gives the following commutative diagram

The following the lemma can be found in[5].

Lemma 4 If C=CRT(C1,C2,···,Cs),then

Lemma 5 Assume the notations given above.A code C=CRT(C1,C2,···,Cs)is a self-dual code over R if and only if the code Ciis a self-dual code over Rifor all 1≤i≤s.

Proof Let C=CRT(C1,C2,···,Cs)is a self-dualcode over R.Suppose anyThen=CRT(C1,···,Ci,···,Cs)by Lemma 4.Therefore ci∈Ci.HenceConversely,for any ci∈Ci,we haveΦ?1(0,···,ci,···,0)∈CRT(C1,···,Ci,···,Cs)= CRTThereforeHenceThis implies that Ciis a self-dualcode over Rifor all1≤i≤s.

Let Cibe a self-dualcode over Rifor all1≤i≤s.Then that C=CRT(C1,C2,···,Cs)is a self-dual code over R follows directly from Lemma 4.

Following these concepts and lemmas above,we have the following theorems.

Proof We first show that the codeis a self-orthogonal code over

Indeed,by assume and Lemma 4,for any i＜∞,we have

By the definition of CRT=Φ?1,

In the following we show thatfor all i＜∞.

This means that[cj,vj]=0.HencesinceΦ?1is an isomorphism for all i＜∞. By the proof of Theorem 2,we know that Cj∞is a self-orthogonalcode over R∞.

Obviously,it is easy to prove that for every k/=j,is a self-orthogonalcodes over

Accounting for all the above,we have

According to the condition of are and Lemma 5,we know thatall self-dual codes overrespectively.By the Theorem 2,we giveis a self-dualcode over R∞.Again to use the Lemma 5,we obtain thatis a self-dualcode over

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tion:94B05,13A99

1002–0462(2014)01–0030–09

Chin.Quart.J.of Math. 2014,29(1):30—38

date:2011-12-05

Supported by the Scientific Research Foundation of Hubei Provincial Education Department(B2013069);Supported by the National Science Foundation of Hubei Polytechnic University(12xjz14A, 11yjz37B)

Biography:LIU Xiu-sheng(1960-),male,native of Huangshi,Hubei,a professor of Hubei Polytechnic University,engages in algebra coding.

CLC number:O157.4 Document code:A