A Transformation Preserving the Dimension and Minimum Weight of Cyclic Codes

Cyclic codes are an important subclass of linear codes and have been extensively studied due to their wide applications. Many classes of optimal ternary and quinary cyclic codes were constructed in the recent literature. However, some cyclic codes of different generator polynomials may have the same optimality. In this paper, we investigate when the cyclic codes of the same length have the same optimality. We consider a transformation preserving the dimension and minimum weight between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-ary cyclic codes of the same length. In addition, three optimal quinary cyclic codes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{(1,e,t)}$\end{document} with parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[5<^>m-1,5<^>m-2m-2,4]$\end{document} are presented, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t=(5<^>m-1)/4$\end{document}.