## 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

*-tuples over*n

*. A linear code*x

*-dimensional subspace of*k

## 2. Algebraic geometric codes

Let

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.

* 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

### 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

* 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

* Proof.* Let us assume

If

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

Observe that

Let

* Proof.* Consider the map

where * j*th elementary symmetric function in the variables

If we apply Theorem 2.7 to the

In

Instead of considering

One can study the orbits of

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

such that

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

and the corresponding triple of partitions

* Proof.* The cyclic code generated by

* Proof* Let

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

The automorphism group of the projective line

Consider the normal rational curve over

is a (* q* + 1)-arc in the

*-dimensional projective space*n

We see that if * q*-dimensional projective subspace, that is, a

*-space. If*q

*+ 1)-arc. It contains*q

*+ 1 points, and every set of*q

### 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).

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