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

04/28/2019
by   Nupur Patanker, et al.
IISER Bhopal
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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

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

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 -Extensions of rational function field

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

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

  7. The pole divisor of is and the pole divisor of is .

  8. Let . Then, the elements with

    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

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

Remark 2.2.

.

2.3. Generalized Hamming weight of Linear codes

The support of a linear code over is defined by

For , the th generalized Hamming weight of is defined by

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

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 , ,

3. Goppa code over Elementary Abelian -extensions of

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

where

.

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

3.1. Parameters of

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

Then, . From section 2, we have:

Hence


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

    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. ),

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

  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

    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

    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

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.

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.

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

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

Then, from [8], Lemma and Lemma ,

where

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], ).

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

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

Proof.

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 ,

Example 4.6.

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

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

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

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,