Counting Irreducible Polynomials of Degree r over Fqn and Generating Goppa Codes Using the Latt

Abstract

The problem of finding the number of irreducible monic polynomials of degreeroverFqnis considered in this paper. By considering the fact that an irreducible polynomial of degreeroverFqnhas a root in a subfieldFqsofFqnrif 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 ofFqnr. We also use the lattice of subfields ofFqnrto determine if it is possible to generate a Goppa code using an element lying in a proper subfield ofFqnr.

The number of extended irreducible binary Goppa codes of degree (2ℓ)m and length 2n+1

“…(2 n ± 1). To count elements in S, the set of all elements of degree (2 ) m in F 2 (2 ) m n , we use the principle of inclusion-exclusion as explained in [11]. That is we simply exclude elements in the subfields F…”

Counting extended irreducible Goppa codes

“…We use a lattice of subfields, as proposed in [9], to show where elements of S(n, n) lie and find |S(n, n)|. Figure 1 shows a lattice of subfields corresponding to q and n = r. Remark 3.1 The number of elements of degree n over…”

Enumeration of extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$

“…In order to simplify our notation we denote all the factors of the degree 2 m by 2 i for 0 ≤ i ≤ m. Now to find the number of elements in S we use the lattice of subfields of F 2 nM , where M = 2 m as done in [7]. Figure 1 shows the lattice of subfields of F 2 nM .…”