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)=TrFqm\Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x) = \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}$$\end{document} where TrFqm\Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}$$\end{document} is a trace polynomial from a field extension of degree m≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 3$$\end{document} has a better minimum distance than what the Goppa bound d≥2deg(g(x))+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 2\deg (g(x))+1$$\end{document} implies. This result is a significant improvement compared to the minimum distance of Trace Goppa codes over quadratic field extensions (the case m=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = 2$$\end{document}). We present two different techniques to improve the minimum distance bound. For general p we prove that the Goppa code C(L,TrFqm\Fq)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(L, \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}})$$\end{document} is equivalent to another Goppa code C(M, h) where deg(h)>deg(TrFqm\Fq)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg (h) > \deg (\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}})$$\end{document}. For p=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2$$\end{document} we use the fact that the values of TrFqm\Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}$$\end{document} are fixed under q–powers to find several new parity check equations which increase the known distance bounds.