Improving the minimum distance bound of Trace Goppa codes

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