Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Lattice of Subfields of Fqnr

Kondwani Magamba,John A Ryan

doi:10.1155/2014/263179

Abstract

The problem of finding the number of irreducible monic polynomials of degree r over Fqn is considered in this paper. By considering the fact that an irreducible polynomial of degree r over Fqn has a root in a subfield Fqs of Fqnr if and only if (nr/s,r)=1, we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of Fqnr . We also use the lattice of subfields of Fqnr to determine if it is possible to generate a Goppa code using an element lying in a proper subfield of Fqnr.

Highlights

In this paper we consider the problem of finding the number, |Pr|, of monic irreducible polynomials of degree r over the field Fqn, where n is a positive integer and q is the power of a prime number
By considering the fact that an irreducible polynomial of degree r over Fqn has a root in a subfield Fqs of Fqnr if and only if = 1, we show that Gauss’s formula for the number of monic irreducible polynomials can be derived by merely considering the lattice of subfields of Fqnr
We are interested in the number of roots of irreducible polynomials of degree r over Fqn because the problem of counting irreducible Goppa codes of length qn and of degree r depends on this number

Summary

Introduction

In this paper we consider the problem of finding the number, |Pr|, of monic irreducible polynomials of degree r over the field Fqn , where n is a positive integer and q is the power of a prime number. Where d runs over the set of all positive divisors of r including 1 and r and μ(k) is the Mobius function; see [1] It has been shown, see [2], that this number can be found by using only basic facts about finite fields and the Principle of Inclusion-Exclusion. This work seeks to emphasize the simplicity of the method given in [2] by using a lattice of subfields This is done by first of all proving Gauss’s formula using the Principle of Inclusion-Exclusion as was done in [2]. We are interested in the number of roots of irreducible polynomials of degree r over Fqn because the problem of counting irreducible Goppa codes of length qn and of degree r depends on this number

Preliminaries

Proof of Gauss’s Formula

Figure 3

Applications to Goppa Codes

Conclusion