On the minimum length of some linear codes

We determine the minimum length nq (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that nq (k, d) = gq (k, d) + 1 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{k-1}-2q^{\frac{k-1}{2}}-q+1 \le d \le q^{k-1}-2q^{\frac{k-1}{2}}$$\end{document} when k is odd, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q^{k-1}-q^{\frac{k}{2}}-q^{{\frac{k}{2}}-1} -q+1 \le d \le q^{k-1}-q^{\frac{k}{2}}-q^{{\frac{k}{2}}-1}$$\end{document} when k is even, and for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2q^{k-1}-2q^{k-2}-q^2-q+1 \le d \le 2q^{k-1}-2q^{k-2}-q^2$$\end{document}.