Improving the minimum distance bound of Trace Goppa codes

01/11/2022

∙

by   Isabel Byrne, et al.

∙

0

∙

share

In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form g(x) = x + x^q + ⋯ + x^q^m-1 where m ≥ 3 (i.e. g(x) is a trace polynomial from a field extension of degree m ≥ 3) has a better minimum distance than what the Goppa bound d ≥ 2deg(g(x))+1 implies. Our improvement is based on finding another Goppa polynomial h such that C(L,g) = C(M, h) but deg(h) > deg(g). This is a significant improvement over Trace Goppa codes over quadratic field extensions (i.e. the case m = 2), as the Goppa bound for the quadratic case is sharp.