Improving the minimum distance bound of Trace Goppa codes

In this paper we prove that the class of Goppa codes whose Goppa polynomial is of the form g(x) = Tr-Fqm (\Fq )where Tr-Fqm \Fq is a trace polynomial from a field extension of degree m = 3 has a better minimum distance than what the Goppa bound d = 2 deg(g(x)) + 1 implies. This result is a significant improvement compared to the minimum distance of Trace Goppa codes over quadratic field extensions (the case m = 2). We present two different techniques to improve the minimum distance bound. For general p we prove that the Goppa code C(L, Tr-Fqm \Fq) is equivalent to another Goppa code C(M, h) where deg(h) > deg(Tr-Fqm \Fq ). For p = 2 we use the fact that the values of Tr-Fqm \Fq are fixed under q-powers to find several new parity check equations which increase the known distance bounds.