True Dimension of Some Binary Quadratic Trace Goppa Codes

We compute in this paper the true dimension over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{F}_2$$ \end{document} of Goppa Codes Γ(L, g) defined by the polynomial \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$g(z) = {\text{Tr}}_{\mathbb{F}_{2^{2s} } :\mathbb{F}_{2^s } } (z)$$ \end{document} proving, this way, a conjecture stated in [14,16].