A class of BCH codes with length q2m-1q+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{q^{2m}-1}{q+1}$$\end{document}.

As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length q2m-1q+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{q^{2m}-1}{q+1}$$\end{document} over the finite fields Fq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {F}}_q$$\end{document}. We settle a conjecture about the largest <italic>q</italic>-cyclotomic coset leader modulo <italic>n</italic> which was proposed by Wu et al. We also get the second largest <italic>q</italic>-cyclotomic coset leader modulo <italic>n</italic> if <italic>m</italic> is odd. Moreover, we investigate the parameters of C(n,q,δi)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta _i)}$$\end{document} (C(n,q,δi)⊥\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta _i)}^\perp$$\end{document}) for i=1,2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$i=1,2$$\end{document} and C(n,q,δ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta )}$$\end{document} (C(n,q,δ)⊥\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}_{(n,q,\delta )}^\perp$$\end{document}) for 2≤δ≤q-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2\le \delta \le q-1$$\end{document}. What’s more, we obtain many (almost) optimal codes. [ABSTRACT FROM AUTHOR]