# A Short Note on Self-Duality of Goppa Codes on Elliptic and Hyperelliptic Function Fields

In this note, we investigate Goppa codes which are constructed by means of Elliptic function field and Hyperelliptic function field. We also give a simple criterion for self-duality of these codes.

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## 1. Introduction

A linear code is a subspace of the

-dimensional standard vector space

over a finite field . Such codes are used for transmission of information. A linear code is called self-dual if , where is the dual of with repect to Euclidean scalar product on . Self-dual codes are an important class of linear codes.

It was observed by Goppa in 1975 that we can use algebraic function fields over to construct a class of linear codes by choosing a divisor and some rational places of algebraic function field over . In this note, we investigate codes which are constructed by means of elliptic and hyperelliptic function fields. This class of codes provide non-trivial examples of geometric Goppa codes.

For self-dual geometric Goppa codes, Driencourt [2] and Stichtenoth [6] showed a criterion, which is too complex to apply. In [4], Xing gave a simple criterion for self-duality of Goppa codes over elliptic function field with base field of characteristic 2. There are some difficulties in generalising the Xing’s criterion to the field of characteristic not equal to . Using Xing’s idea and results from [3], we give a simple criterion for self-duality of Goppa codes over elliptic function field and hyperelliptic function field of characteristic not equal to .

## 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 geometric Goppa code associated with and is defined by

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

Then, is an code with parameters and .

Another code can be associated 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

The dual code of is i.e. Let be a Weil differential such that and for , then .

### 2.2. Elliptic Function Field

###### Definition 2.1.

An algebraic function field (where is the full constant field of ) is said to be an elliptic function field if the following conditions hold:

• the genus of is , and

• there exists a divisor with .

The following theorems characterize elliptic function field over (where ).

###### Theorem 2.2 ([5], Chapter VI).

Let be an elliptic function field. If , there exist such that and

 y2=f(x)∈K[x]

with a square-free polynomial of degree 3.

###### Theorem 2.3 ([5], Chapter VI).

Suppose that with

 y2=f(x)∈K[x]

where is a square-free polynomial of degree 3. Consider the decomposition of into monic irreducible polynomials with . Denote by the place of corresponding to , and by the pole of . Then the following holds:

1. is the full constant field of , and is an elliptic function field.

2. The extension is cyclic of degree 2. The places and are ramified in ; each of them has exactly one extension in , say and , and we have , and .

3. and are the only places of which are ramified in , and the different of is

 Diff(F/K(x))=Q1+⋯+Qr+Q∞.

### 2.3. Hyperelliptic function field

###### Definition 2.4.

A hyperelliptic function field over is an algebraic function field of genus which contains a rational subfield with .

###### Lemma 2.5 ([5], Chapter VI).

Assume that .

1. Let be a hyperelliptic function field of genus . Then there exist such that and

 (2.1) y2=f(x)∈K[x]

with a square-free polynomial of degree or .

2. Conversely, if and with a square-free polynomial of degree , then is hyperelliptic of genus

3. Let with as in . Then the place which ramify in are the following:

all zeros of if ,
all zeros of and the pole of if .

## 3. Self-duality of Geometric Goppa codes over Elliptic Function Field F/k with char K≠2

In [4], Xing gave a criterion for self-duality of Goppa codes over elliptic function field with base field of characteristic To get the criteria he has used the following proposition:

###### Proposition 3.1.

Let , and be two positive divisors such that

1. , and .

2. and for any .

3. .

Then if and only if .

This proposition doesn’t apply when and are not positive divisors (For example: Let be an algebraic function field, if be a place of such that its extension in has degree , then for and we have but ). In this case, we observed that we can use following theorem to determine the self-duality of Goppa codes over elliptic and hyperelliptic function fields.

###### Definition 3.2.

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

###### Theorem 3.3 ([3], Corollary 4.15).

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

We start with , large enough and characteristic of . Let be an elliptic function field. Then, with such that

 y2=f(x)∈K[x]

with a square-free polynomial of degree 3.

Consider the decomposition of into monic irreducible polynomials with . Denote by the place of corresponding to , and by the pole of .

Let an even positive integer. Let be places of degree 1 of such that

• For each , has exactly two extensions in say, and .

Let . Let such that in . Let be a divisor of of degree . Let be a differential in defined by

 η=g′(x)dxg(x).

Then, we get and for all . Therefore, .
Now, . Then the condition for self-duality of is given by the following theorem.

###### Theorem 3.4.

With all conditions as above, is self-dual if and only if for some such that for .

###### Proof.

is self-dual iff .
By [3] corollary 4.15,

 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) ⇔ (g′(x))=2G−(u)−(n−3)Q∞−Q1−⋯−Qr

## 4. Self-duality of geometric Goppa codes over Hyperelliptic function field F/k of genus 2 with char K≠2

Let , large enough and characteristic of . Let be a hyperelliptic function field of genus 2. Then there exist such that and

 y2=f(x)∈K[x]

with a square-free polynomial of degree 5. Then the places corresponding to all zeroes of and the pole of ramify in .

Let be an even positive integer. Let be zeros of and pole of . Therefore, ramify in . Let be places of degree 1 of such that

• For each , has exactly two extensions in say, and .

Let . Let such that in . Let be a divisor of of degree . Let be a differential in defined by

 η=g′(x)dxg(x).

Then, we get and for all . Therefore, .
Now, . Then the condition for self-duality of is given by the following theorem.

###### Theorem 4.1.

With all conditions as above, is self-dual if and only if for some such that for .

###### Proof.

is self-dual iff .
By [3] corollary 4.15,

 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) ⇔ (g′(x))=2G−(u)−(n−3)Q∞−Q1−⋯−Qr.

## 5. Concluding Remarks

In this note, we have investigated Goppa codes over Elliptic and Hyperelliptic function fields with base field of characteristic not equal to . We gave a simple criterion for self-duality of these codes.

## References

• [1] Arnaldo Garcia, On goppa codes and artin-schreier extensions, Communications in Algebra, Volume 20,1992-Issue 12, Pages 3683-3689.
• [2] Yves Driencourt and Henning Stichtenoth, A criterion for self-duality of geometric codes, Communications in Algebra, Volume 17, 1989 - Issue 4.
• [3] Carlos Munuera and Ruud Pellikaan, Equality of geometric Goppa codes and equivalence of divisors, Journal of Pure and Applied Algebra, Volume 90, Issue 3, 13 December 1993, Pages 229-252.
• [4] X. Chao-Ping, Remarks on self-dual elliptic codes, Chinese Science Bulletin 36(8), 629-631(1991).
• [5] Henning Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag Berlin Heidelberg, 1993.
• [6] Henning Stichtenoth, Self-dual Goppa codes, Journal of Pure and Applied Algebra Volume 55, Issues 1–2, November 1988, Pages 199-211.