The Classification of Some Perfect Codes

Perfect 1-error correcting codes C in Z2n, where n=2m−1, are considered. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \left\langle C \right\rangle $$ \end{document}; denote the linear span of the words of C and let the rank of C be the dimension of the vector space\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \left\langle C \right\rangle $$ \end{document}. It is shown that if the rank of C is n−m+2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n−m+2, described by a Hamming code H and a set of MDS-codes Dh, h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \in $$ \end{document}H, over an alphabet with four symbols. The case of rank n−m+1 is much simpler: Any such code is a Vasil'ev code.