A new construction of Algebraic Geometry code using Trace function

02/29/2020 ∙ by Nupur Patanker, et al. ∙ IISER Bhopal 0

In this note, we give a construction of Algebraic-Geometry codes on algebraic function field F/ F_q using places of F (not necessarily of degree one) and trace functions from various extensions of F_q. We compute a bound on the dimension of this code. We also determine a bound on the minimum distance of this code in terms of B_r(F) ( the number of places of degree r in F), 1 ≤ r < ∞. This code is a generalization of the geometric Goppa code, with no restriction on the length of the code except the support condition on divisors defining the code. We obtained few quasi-cyclic codes over F_p as examples of these codes.

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1. Introduction

A linear code is a -subspace of

, the n-dimensional standard vector space over a finite field

. Such codes are used for transmission of information. 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 .

A very useful method of constructing codes over is to use the trace mapping , if a code over is given. The codes so obtained are called trace codes. Many interesting codes over can be represented as trace codes. Trace code, its dimension and Hamming weigth have been studied in [1], [3], [4], [5], [6], [7], [8], [11], etc. Generalized hamming weights of trace codes have been studied in [9] and [10]. In [8]

, Conny Voss obtained an estimate for the weights of code words of trace codes by using the Hasse-Weil bound for the number of rational places over

. In this note, we give a construction of Algebraic-Geometry codes on algebraic function field using places of (not necessarily of degree one) and trace functions from various extensions of . This code is a generalization of the geometric Goppa code, with no restriction on the length of the code except the support condition on divisors defining the code. By using ideas from [8], we compute a bound on the dimension of this code. We also determine a bound on the minimum distance of this code in terms of  ( the number of places of degree in ), .

A linear code of length over is called a quasi-cyclic code of index if for every we have . Quasi-cyclic codes have been studied for many years. The algebraic structure has been studied in [18], [19], [20], [21], etc. In [17], the authors have studied geometric realisation of quasi-cyclic codes. In this note, we construct few quasi-cyclic codes over , where is prime, as examples of .

This note is organized as follows. In section , we recall some results about Goppa codes, Extensions of Algebraic function fields and Trace codes. In section , we give the definition of code . In section , we determine an upper bound on the dimension of code . In section , we determine lower bound on the minimum distance of code under various conditions. In section , we conclude the note listing examples of quasi-cyclic code over derived from the code .

2. Preliminaries

2.1. Goppa code

[11], Chapter Goppa’s construction of linear code 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


Then, is an code with parameters and .

2.2. Extensions of Algebraic Function Fields

Let denotes an algebraic function field of one variable with full constant field . The field is assumed to be perfect.

Definition 2.1.

[11], Chapter An algebraic function field is called an algebraic extension of if is an algebraic field extension and The algebraic extension of is called a finite extension if .

Definition 2.2.

[11], Chapter A place is said to be an extension of if .

Definition 2.3.

[11], Chapter Let be a subring of . An element is said to be integral over if for some monic polynomial , i.e. if there are such that

The next theorem describes a method which can often be used to determine all extensions of a place in . For convenience we introduce some notation.
is the residue class field of
is the residue class of
If is a polynomial with coefficients , we set

Theorem 2.4.

(Kummer) [11], Suppose that where is integral over , and consider the minimal polynomial of over . Let

be the decomposition of into irreducible factors over . Choose monic polynomials with

Then for , there are places satisfying

Moreover , for .
Under additional assumptions one can prove more: Suppose that atleast one of the following hypotheses is satisfied:


Then there exists, for , exactly one place with and . The places are all places of lying over , and we have . The residue class field is isomorphic to , hence .

The next theorem we need gives an estimate for the number of places of a fixed degree . Given a function field of genus , we define

Theorem 2.5.

[11], The estimate

holds for all .

2.3. ( [11], VIII.1) Trace codes

Consider the field extension , let denote the trace mapping. For , we define .

Definition 2.6.

( [11], Def. VIII.1.1) Let be a code over . Then

is called the trace code of .

A subcode of a code means an -subspace . By we denote the set

A bound on the dimension of the trace code is given by the following proposition.

Proposition 2.7.

( [11], Theorem VIII.1.4) Let be a code over and be a subcode with the additional property . Then

3. Definition of code

Let be an algebraic function field. Choose a positive integer . Choose arbitrary places of . Let . Let . Choose a divisor of with . Let denote the trace map from to .
Consider the map

Define . Then, is a linear code of length over .

Remark 3.1.

There is no restriction on the length of the code except its support doesn’t intersect with .

4. Dimension of code

We have is a vector space over . Hence,

where is the kernel of . Note that . Let

For any we have . Therefore, the -vector space generated by denoted by is contained in and

Now, for a divisor we define the divisor , where denotes the greatest integer function. For , we have and . Consider the map

The kernel of the map is .Also .

From the above discussion, we have the following result regarding the dimension of ,

Theorem 4.1.

.

5. Minimum distance of code

In this section we apply the techniques of [11], VIII. to compute bound on the minimum distance of . Suppose where . We first assume that for all , is coprime to and i.e degrees of are not times multiples of each other.

Given , there exists a unique effective divisor of smallest degree such that for all . One can describe as follows: choose a basis of then

for all . We associate with the divisor a second divisor defined by

Definition 5.1.

An element is called degenerate if where , and . Otherwise, is said to be non-degenerate. A -subspace of is said to non-degenerate if every element of is non-degenerate.

Remark 5.2.

Assume that is non-degenerate.

For , consider the field extension defined by . Consider the polynomial . Since, is non-degenerate by [2] Lemma , we have is irreducible over . Thus, is a Galois extension of degree . Also,

Lemma 5.3.

is constant field of .

Proof.

Let be the constant field of . By definition of constant field extension,

So, by Galois correspondence, there exists a field extension of such that . Thus, by Artin-Scherier extension, where . Since, . By Elementary Abelian -Extension, where . and are two generators of Artin extension. So, they are related as for and . This implies, . This is a contradiction as is assumed to be non-degenerate. ∎

5.1. Case I: All have same degree

The support of consists of two disjoint subsets where

and

Hilbert’s Theorem 90 states that for ,


We would like to determine how the places decompose in the extension .

Let and . The Artin-Scherier polynomial has no root in . Each irreducible factor of in has degree . So, by Kummer’s theorem there are no places of degree in over in .
Next we consider a place . Then can be written as , hence the polynomial factors over into distinct places of , all of degree .
The above considerations imply that

Therefore, Applying Theorem , we get

Theorem 5.4.

If and , then the minimum distance of is bounded from below by


5.2. Case II: Degrees of are not same

Let be the set of distinct ’s ( where as defined before). For , let

and

Let

For , let and . Then as above, for

Therefore,

Theorem 5.5.

If and , then the minimum distance of is bounded from below by


Now we consider the case when degrees of are times multiple of each other.
 
Suppose and be the set of distinct ’s such that for and , . For and , let

and

Let


Let then for some such that and . The polynomial has no root in . It decomposes into places of degree . Now for some with , by Kummer’s theorem has extensions all of degree .

Let and and and .

The above considerations imply that

Applying Theorem for we get,

Now, . Proceeding as above, we get

Theorem 5.6.

If and , then the minimum distance of is bounded from below by

6. Examples

6.1. Trace code

For , if , then we get the code is the trace code of geometric goppa code.

6.2. Some examples of quasi-cyclic codes over

We let where is prime. For a positive integer it is well known that the number of monic irreducible polynomials of degree over , denoted by , is given by

(6.1)

The trace of a polynomial of degree over is the coefficient of in .

For , let denotes the number of monic irreducible polynomials over of degree and trace . Carlitz’s formula [12] for when is

Lemma 6.1.

[13], Lemma If and are non-zero elements of then .

Proposition 6.2.

[13], Corollary Write with . Then

where if and if .

Proposition 6.3.

[14],Theorem Let be a subfield of two isomorphic finite fields and . If is a field isomorphism from onto that fixes the elements of , then for ,

Fix a prime and positive integers and such that . Consider the finite field where is a prime. Then from Lemma and Proposition , we have for any we have .

Let denotes the trace function

Choose such that mod or in other words (such a always exist since ). Let denotes the set of all monic irreducible polynomials of degree over . We know that automorphism group of rational function field is the projective linear group over i.e any is of the form

Let given by .

Since for any i.e for monic irreducible polynomial of degree , we have is monic and irreducible of degree i.e and conversely. Order the elements of as

where for , the -th row consists of elements of with trace and for ,   . Then the corresponding set of places of is

Choose a positive integer . Then the code with and is

(6.2)
Proposition 6.4.

The code as in equation is quasi-cyclic code of length and index .

Proof.

We have for as defined before and . Let be defined by , for and . We extend the action of to as

(6.3)

We claim that .
 
We have the following diagram,

where