## Abstract

We study and enumerate cyclic codes which include generalised Reed-Solomon codes as function field codes. This geometrical approach allows to construct longer codes and to get more information on the parameters defining the codes. We provide a closed formula in terms of Stirling numbers for the number of irreducible polynomials and we relate it with other formulas existing in the literature. Further, we study quasi-cyclic codes as orbit codes in the Grassmannian parameterizing constant dimension codes. In addition, we review Horn’s algorithm and apply it to construct classical codes by their defining ideals.

### Keywords

- cyclic code
- partition
- Grassmannian

## 1. Introduction

Function fields are used ubiquitously in algebraic coding theory for their flexibility in constructions and have produced excellent linear codes. Suitable families of function fields, for example good towers of function fields, have been used to construct families of codes with parameters bound better than the asymptotic bound.

Let

The encoding of an information word into a *k*-dimensional subspace is usually known as coding for errors and erasures in random network coding [1]. Namely, let

Let

It is a polynomial in

The number of points

Garcia and Stichtenoth analysed the asymptotic behaviour of the number of rational places and the genus in towers of function fields, [2]. From Garcia-Stichtenoth’s second tower one obtains codes over any field

One of the main problems in coding theory is to obtain non-trivial lower bounds of the number

**Notation.** Let *n*-tuples over *x*. A linear code *k*-dimensional subspace of

## 2. Algebraic geometric codes

Let

**Definition 2.1.** *Algebraic Geometric Codes (AGC) are constructed by evaluation of the global sections of a line bundle or a vector bundle on the curve**over**distinct rational places**. Namely, let**be the function field of the curve,**the divisor**and**a divisor of**of degree**such that**. Then the geometric Goppa code associated with the divisors**and**is defined by*

Recall that

In the sequel, an *k*-dimensional subspace of

### 2.1 Generalised Reed-Solomon codes as cyclic codes

Another important family of Goppa codes is obtained considering the normal rational curve (NRC)

Assuming that

is injective, since the existence of a non-zero polynomial of degree less than

**Theorem 2.2**. *Assume that**, then the number of polynomials of degree**decomposable into distinct linear factors over a finite field**of arbitrary characteristic a prime number**, is equal to**, where**is the falling factorial polynomial**, where**is the Stirling number of the first kind (the number of ways to partition a set of**objects into**non-empty subsets), divided by the order of the affine transformation group of the affine line**, that is*

*Proof.* We need to count all the polynomials

Theorem 2.3. *Given a set of integers**module**, there is a set**of**integers which is a set of roots, that is, there is a polynomial**, where**is a generator of**for some prime number**and**is the least integer such that**. The ideal**generates in**is a cyclic linear code of parameters*

*Proof.* Let

As an application of Theorem 2.2, given an integer

In the theory of error-correcting codes to a given code

**Definition 2.4**. *The distance between vectors**and**in the Weighted Hamming metric (WHM) is defined by a function:*

where

Geometrically a binary vector

**Example 1.** *Consider the Goppa code defined by the rational function**which admits as decomposition into partial fractions the expression**. The presence of a double factor**corresponds to the existence of an eigenspace**in the vector space**of multiplicity 2 and thus an**splitting subspace where the operator**is just the linear operator**, with**the eigenvalue associated to**and**is the generator matrix of the code. We recall that an**dimensional**subspace is**splitting if**is invariant under the action of any element**in the Galois group of the extension*

### 2.2 Algebraic function field codes

A much greater variety of linear codes is obtained if one uses places of arbitrary degree rather than just places of degree 1. These codes are more naturally described through function field codes. A general viewpoint is that function field codes are certain finite dimensional linear subspaces of an algebraic function field over a finite field as in Goppa’s construction.

In the paper [5], the authors introduce another construction where places of arbitrary degree are allowed. The method consists of choosing two divisors

**Example 2.** *We consider as in* [7] *the Suzuki curve**defined over**by the following equation**with**and**. This curve has exactly**rational places with a single place at infinity**and it is of genus**We construct a code out of the divisor* *and**where**is the sum of the**rational points and the parameter**satisfies the bound**and**is the genus of the curve.*

Observe that the geometric Goppa code

where

**Theorem 2.5.** *Cyclic codes are function field codes constructed over the curve**with affine equation**defined over a finite field**of**elements, where**is a power of a prime**and**are integer numbers greater or equal than 2.*

*Proof.* Let us assume

If

**Remark 2.6.** *The proof given in theorem gives a realisation of cyclic codes as AG codes constructed over the curve with affine model**. In particular when**, we cover the codes defined over the Hermitian curve.*

Another important family of cyclic codes is obtained considering the roots of the polynomial

**Example 3.** *The polynomial**over**is irreducible, thus the fields**are isomorphic, and the roots**correspond to one place of degree 2 in the extension field*

**Example 4.** *We define the polynomials**, with**, and**. Then, if**is a separable polynomial, then the Galois group of**over**is a regular extension with Galois group*

Observe that

**Theorem 2.7.** *(Hilbert) Let**be acting on**. The field**of**invariants is**, where**is the**symmetric polynomial in**and**has Galois group**over**. It is the splitting field of the polynomial*

Let

**Proposition 2.8.** *Let**be a factorisation of an integer positive number**into irreducible coprime factors and assume**, then there is a sequence of field extensions*

*Proof.* Consider the map

where *j*th elementary symmetric function in the variables

If we apply Theorem 2.7 to the

**Example 5.** *Assume**and we study again the roots of the polynomial**in its splitting field. Let**be a non-trivial n-root of unity, for any divisor**of**, one can consider the symbols**. The field of**invariants of the polynomial**is the set of solutions to the equation:*

In

Instead of considering

One can study the orbits of

**Example 6.** *We study the action of a rotation element on the Grassmannian**of lines in a 3-dimensional projective space**. We apply to any line**a rotation**of angle**, represented by the array of vectors**. It is easy to see that the orbit code by the composed action**with**divisor of**is a quasi-cyclic code of index*

In general, we study generalised Grassmannians or more commonly known as flag varieties. Fix a partition

such that

**Proposition 2.9.** *Let us consider the factorisation of**into irreducible pairwise coprime factors**with**, and**be the partition of exponents. Then there is a flag variety**of partial flags of**vector spaces:*

such that

*Proof* Observe that the result follows trivially for the case in which

Given a cyclic code over

Let

By Möebius inversion:

In the case of binary codes where

where

Let

The code

Given a triple

We describe Horn’s inductive procedure to produce set of triples

**Example 7.** *Let us consider the triple of subsets*

and the corresponding triple of partitions

**Lemma 2.10.** *For any triple**admissible for the Horn problem, the polynomials defined by**and**generate a cyclic code of length**and**, where**and**is the characteristic of the field*

*Proof.* The cyclic code generated by

**Remark 2.11.** *We see that Horn’s algorithm is relevant since some classical code constructions can be seen as ideals in a finite dimensional commutative semi simple algebra over a finite field**with**elements and**a prime number as in example (3).*

**Lemma 2.12.** *The family of cyclic codes obtained by considering the roots of the polynomial**over its splitting fields are indeed AG codes arising from genus 0 curves, and by Riemann-Roch theorem, their parameters satisfy the bound**, where**is the minimum distance.*

*Proof* Let

### 2.3 Generating functions of conjugacy classes in a group

The automorphism group of the projective line

**Definition 2.13.** *In**a**arc is a set of**points any**of which form a basis for**, or in other words,**of them but not**are collinear.*

Consider the normal rational curve over

is a (*q* + 1)-arc in the *n*-dimensional projective space

We see that if *q*-dimensional projective subspace, that is, a *q*-space. If *q* + 1)-arc. It contains *q* + 1 points, and every set of

### 2.4 Conclusion

The problem of considering finite subgroups and conjugacy classes in

## Acknowledgments

This research has been partially supported by the COST Action IC1104 and the project ARES (Team for Advanced Research on Information Security and Privacy. Funded by Ministry of Economy and Competitivity).