Minimal codewords over finite fields derived from certain graphs

Throughout this paper, we explore the number of non-equivalent minimal codewords of linear codes derived from certain graphs. We propose a lower bound on the number of non-equivalent minimal codewords over 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} associated with graphs of diameter 2. Beyond diameter 2, we also determine the number of non-equivalent minimal codewords over 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} for graphs with arbitrary diameter. To achieve this, we study n-cycles and the row spaces generated by some rows from the generator matrix of linear codes. Primarily, our focus is on the number of non-equivalent minimal codewords, and we also provide precise construction methods for identifying minimal codewords in linear codes. To support our results, we present some examples in this work.