# A short note on Goppa codes over Elementary Abelian p-Extensions of F_p^s(x)

In this note, we investigate Goppa codes which are constructed by means of Elementary Abelian p-Extensions of F_p^s (x), where p is a prime number and s is a positive integer. We give a simple criterion for self-duality of these codes and list the second generalized Hamming weight of these codes.

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## 1. Introduction

Let be the finite field with elements of characteristic (where is a positive integer). A linear code is a -subspace of , the

-dimensional standard vector space over

. Such codes are used for transmission of information. It was observed by Goppa in 1975 that we can use algebraic function field over to construct a class of linear codes. In Goppa’s construction, we choose a divisor and rational places of the algebraic function field to form a linear code of length . In this note, we study Goppa codes over Elementary Abelian -Extensions of .

The properties of Elementary Abelian -Extensions of have been studied in [2], [1], etc. In [11], T. Johnsen, S. Manshadi and N. Monzavi determined parameters of Goppa codes over plane projective curves with affine equation , where is a separable, additive polynomial of degree , for some and the degree of is not divisible by . They studied codes over with the assumption that . In [3], Garcia also studied Goppa codes over plane projective curves with affine equation but with a central hypothesis that there exists a subgroup of such that if with , then . In this note, we investigate codes which are constructed by means of Elementary Abelian -Extensions of without the above assumptions. We determine a simple condition for self-duality of these codes and list their second generalized Hamming weights.

This note is organised as follows. In section , we recall some results about Goppa’s construction of linear codes, Elementary Abelian -Extensions of and generalized Hamming weight of linear codes. In section , we study the properties of one-point codes over this function field. In section , we determine a simple condition for self-duality of these codes. In section , we conclude the note listing the second generalized Hamming weights of these codes.

## 2. Preliminaries

### 2.1. Goppa code

Goppa’s construction is described as follows:

Let be an algebraic function field of genus . Let be pairwise distinct places of of degree 1. Let and be a divisor of such that . The Goppa code associated with and is defined as

 CL(D,G):={(x(P1),⋯,x(Pn)): x∈L(G)}⊆Fnps.

Then, is an code with parameters and .

We can define another code with the divisors and by using local components of Weil differentials. We define the code by

 CΩ(D,G):={(ωP1(1),⋯,ωPn(1)): ω∈ΩF(G−D)}.

Then, is an code with parameters and

is the dual code of with respect to Euclidean scalar product on i.e. Let be a Weil differential of such that and for . Then, .

### 2.2. ([7], p.200) Elementary Abelian p-Extensions of rational function field

Let be a field of characteristic . Consider a function field with

 yq+μy=f(x)∈K[x],

where is a power of and . Assume that is coprime to . Also assume that all the roots of the equation are in . Then the following holds:

1. , and is the full constant field of .

2. is Galois. The set is a subgroup of order of the additive group of .

3. The pole of in has a unique extension , and . Hence is a place of of degree one.

4. is the only place of which ramifies in .

5. The genus of is .

6. The divisor of the differential is

 (dx)=(2g−2)Q∞=((q−1)(m−1)−2)Q∞.
7. The pole divisor of is and the pole divisor of is .

8. Let . Then, the elements with

 0≤i, 0≤j≤q−1, qi+mj≤r

form a basis of the space over .

9. For all , one of the following cases holds:
Case (1). The equation has distinct roots in . In this case, for each with there exists a unique place such that and . Hence, has distinct extensions in , each of degree one.
Case (2). The equation has no root in . In this case, all extensions of in have degree .

###### Remark 2.1.

The Hermitian function field over is defined by

 H=Fq2(x,y)  with  yq+y=xq+1.

This is a special case of Elementary Abelian -Extension with , and .

.

### 2.3. Generalized Hamming weight of Linear codes

The support of a linear code over is defined by

 supp(C):={i : xi≠0 for some x=(x1,⋯,xn)∈C}.

For , the th generalized Hamming weight of is defined by

 dl(C):=min{ ∣supp(D)∣ : D is a linear subcode of C with dim(D)=l}

In particular, the first generalized Hamming weight of is the usual minimum distance. The weight hierarchy of the code is the set of generalized Hamming weights . These notions of generalized Hamming weights for linear codes were introduced by Wei in his paper [12].

Few properties of generalized Hamming weight of have been listed in the following theorems.

###### Theorem 2.3.

[12]Monotonicity For an linear code with , we have

 1≤d1(C)

Let be a parity check matrix of , and let , , be its column vectors. For , let denote the space generated by those vectors. Then

###### Theorem 2.4.

[12]

For Goppa code , the th generalized Hamming weight is given by the following theorem.

###### Theorem 2.5.

[5] Let be a code of dimension and . Then for every , ,

 dl(C) =min{deg(D′) : 0≤D′≤D, dim(L(G−D+D′))≥l+a} =min{n−deg(D′) : 0≤D′≤D, dim(L(G−D′))≥l+a}.

## 3. Goppa code over Elementary Abelian p-extensions of Fps(x)

Let be a rational place in . A positive integer is called pole number at is there exists such that . Let be the sequence of pole numbers at (that is, is the th pole number at ); thus , so .

Few properties of Elementary Abelian -Extensions of , observed from [7] and [10], are listed in the following theorem.

###### Theorem 3.1.

(Properties of function field)

• The Weierstrass semigroup of is generated by and i.e. .

• is symmetric numerical semigroup.

• is the largest gap number of .

• is a canonical divisor.

• If denotes the gap numbers of , then .

• The sequence satisfy the isometry dual condition where .

• The polynomial , where , is absolutely irreducible.

• is an integral basis of for all .

In [7], Stichtenoth has investigated one-point Goppa codes over Hermitian function field. Using similar idea, we define codes over Elementary Abelian -Extension of and determine its parameters.

Let , large enough. Let be an integer coprime to . Choose distinct elements . Let . Denote by the zero of in . Let denotes the zeroes of in (choose large enough so that all the zeroes are in ). Then, for , , are the places of of degree one (such places exist by section ).

###### Definition 3.2.

For r we define

 Cr:=CL(D,rQ∞),

where

 D:=m∑i=1q∑j=1Pαi,βj

.

Then, is a code of length over the field . For , , therefore . For , , therefore . It remains to study codes with .

### 3.1. Parameters of Cr

Let be the set of pole numbers of . For , let

 J(b):={u∈J| u≤b}.

Then, . From section 2, we have:

 J(b)={u≤b| u=iq+jm with i≥0 and 0≤j≤q−1}.

Hence

 |J(b)|=|{(i,j)∈N0×N0; j≤q−1 and iq+jm≤b}|.

###### Theorem 3.3.

[7], is an code with parameters and . If , then .

###### Theorem 3.4.

Suppose that . Then the following holds:

1. . For , . For , we have .

2. The minimum distance of satisfies

 d≥qm−r.

If , where or if , where , then . In addition, if then is not MDS code.

###### Proof.
1. By Theorem and as we have, . For (i.e. ),

 dim(Cr)=dim(L(rQ∞))=|J(r)|.

For (i.e. ), Riemann-Roch theorem yields

 dim(Cr)=dim(L(rQ∞))=deg(rQ∞)+1−g=r+1−(q−1)(m−1)/2.
2. The inequality directly follows from Theorem . If , where , choose distinct elements from the set ( where as before are such that ). Let us call these elements . Then the element

 z1:=b∏j=1(x−γj)∈L(rQ∞)

has exactly distinct zeros in . The weight of the corresponding codeword in is . Hence, .

Similarly, if , where , choose distinct elements from the set . Let us call these elements . Then the element

 z2:=c∏j=1(y−τj)∈L(rQ∞)

has exactly distinct zeros in . The weight of the corresponding codeword in is . Hence, .

If and is MDS code then, implies which is not possible. Similarly for .

Using the idea from [8], we have the following result for minimum distance of .

###### Theorem 3.5.

Assume . For we have . Let be the largest integer such that is a pole number at i.e. where and . Then, the minimum distance of satisfies

 d(Cr)=a+2.
###### Proof.

Let be a parity check matrix of . From section , we have , is a basis for . Choose such that . Let be a submatrix of with columns corresponding to . We write in the following form using row reduction.

 H1=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣111⋯1α1α2α3⋯αa+2α21α22α23⋯α2a+2⋮⋮⋮⋮αa1αa2αa3⋯αaa+2000⋯0⋮⋮⋮⋮000⋯0⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

Here, and has columns, so the columns of are linearly dependent. Therefore, .

On the other hand, we choose any distinct columns from . Let us call this matrix . Since each column of corresponds to a place of degree , we reorder columns of according to as follows.

 Pα1,β1,1,Pα1,β1,2,⋯,Pα1,β1,w1Pα2,β2,1,Pα2,β2,2,⋯,Pα1,β2,w2⋮⋮⋮⋮Pαγ,βγ,1Pαγ,βγ,2⋯,Pαγ,βγ,wγ

where ’s are pairwise distinct and with . For , belongs to basis of . We rewrite these basis elements in the form

 1,y,y2,⋯,yw1−1x,xy,xy2,⋯,xyw2−1x2,x2y,x2y2,⋯,x2yw3−1⋮⋮⋮⋮xγ−1,xγ−1y,xγ−1y2,⋯,xγ−1ywγ−1

Then, we extract an submatrix of such that each row corresponds to a function above in the given order. That is, , where is a matrix with with

 Bi,j=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣111⋯1βj,1βj,2βj,3⋯βj,wjβ2j,1β2j,2β2j,3⋯β2j,wj⋮⋮⋮⋮βwi−1j,1βwi−1j,2βwi−1j,3⋯βwi−1j,wj⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

Then, from [8], Lemma and Lemma ,

 det(H′)=(γ∏i=1det(Bi,i)).(γ∏j=2ρwjj)

where

 ρj=j−1∏i=1(αj−αi), j=2,3,⋯,γ.

And any columns of are linearly independent over . Hence, . ∎

## 4. Condition for self-duality of codes

A linear code is called self-dual if , where is the dual of with respect to Euclidean scalar product on . Self-dual codes are an important class of linear codes. We give a simple criterion for self-duality of codes over Elementary Abelian -Extensions of by using the following theorem from [6].

###### Definition 4.1.

We call two divisors and equivalent with respect to if there exists such that and , for all

###### Theorem 4.2 ([6], 4.15).

Suppose . Let and be two divisors of the same degree on a function field of genus . If is not equal to nor to and , then if and only if and are equivalent with respect to .

Let and as before. Clearly, . Let be a differential then we get, and for all . Therefore, for any divisor on with , we have .

###### Theorem 4.3.

The dual code of is given by

 C⊥r=¯a C2qm−q−m−1−r

where . Hence, if then is quasi-self-dual if and only if .

###### Proof.
 C⊥r =CL(D,D+(η)−rQ∞) =CL(D,D+(f′(x))−(f(x))+(dx)−rQ∞) =CL(D,D+(f′(x))−D+qmQ∞+(2g−2)Q∞−rQ∞) =CL(D,(f′(x))+(qm+(2g−2−r)Q∞) =¯aCL(D,(2qm−q−m−1−r)Q∞) =¯aC2qm−q−m−1−r

Now it follows directly from Theorem that is quasi-self-dual if and only if . ∎

###### Corollary 4.4.

For , define . The dimension of is given by

Let be a divisor of with . Clearly, . Let . Then, . From Theorem , it follows that

###### Theorem 4.5.

is self-dual if and only if is equivalent to with respect to .

###### Proof.

By Theorem ,

 CL(D,G)=CL(D,D+(η)−G) ⇔ G=D+(η)−G+(u) for some u∈F such that u(P)=1 for each P∈supp(D) ⇔ (u)+(η)=2G−D ⇔ (u)+(f′(x))+(dx)−(f(x))=2G−D ⇔ (u)+(f′(x))+[(q−1)(m−1)−2]Q∞−D+qmQ∞=2G−D ⇔ (f′(x))=2G−(2qm−q−m−1)Q∞−(u).

###### Example 4.6.

Let . Let . Let be a primitive element of . Consider with

 y2+y=x(x−1)(x−ω)

Therefore, all roots of is in . The genus of is . Let

 f(x)=x(x−1)(x−ω).

Let and denotes zero in of and respectively. Then, each of and has exactly two extensions in . Similarly, the zero of denoted by has two extensions in , say, and . Let and let be a divisor in equivalent to with respect to . Then is self dual.
Conversely, if is self-dual code with as above then is equivalent to with respect to .

## 5. Generalized Hamming weight of code Cr

Using the idea of [5], we have the following lemma.

###### Lemma 5.1.

Let be a pole number at . Then, where and . If either or then a divisor such that .

###### Proof.

For and , works.

If and then, . With notation as in section , choose elements from . Denote these elements by . Define . Then, . Therefore, .

Similarly for and . ∎

###### Definition 5.2.

A positive integer is said to have property (*) if is a pole number at , for , and either or .

###### Theorem 5.3.

As before, let . If for , or has the property (*) then, .

###### Proof.

If has the property (*), then according to Lemma , there exists a divisor such that . Thus, . Hence, from Theorem , .

Now, if has the property (*) then according to Lemma there exists a divisor such that . Also, . Thus, . Therefore,