Equivalent Irreducible Goppa Codes and the Precise Number of Quintic Goppa Codes of Length 32

In this paper, by defining an irreducible Goppa code in terms of a single field element, we give a simple proof of the well known result, namely that if one irreducible Goppa polynomial can be transformed into another by a combination of an affine map and a Frobenius automorphism, then the corresponding codes are equivalent. We give the details of the proof for the most general case, namely for irreducible Goppa codes over any finite field and further show that the equivalence in question is one of the simplest types, that is a permutation of coordinate positions. Finally, by giving a full categorisation of irreducible Goppa codes of degree 5 and length 32, we show 1) that contrary to a previous claim, even if two Goppa polynomials are not related by the above two maps, the Goppa codes can still be equivalent and 2) the upper bound on the number of irreducible Goppa codes given in a recent publication is not tight.