1. Some results on the Hamming distances of cyclic codes.
- Author
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Pan, Guantao, Li, Lanqiang, Cao, Ziwen, and Tian, Fuyin
- Abstract
Cyclic codes over finite fields have been studied for decades due to their wide applicability in communication systems, consumer electronics, and data storage systems. Let
p be an odd prime and lets andm be positive integers. In this paper, we first determine the Hamming distances of all cyclic codes of length 8 over Fq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_q$$\end{document}. Building upon this, we explicitly obtain the Hamming distances of all repeated-root cyclic codes of length 8ps\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$8p^s$$\end{document} over Fq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F_q$$\end{document}. As an application, we give all maximum distance separable cyclic codes of length 8ps\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$8p^s$$\end{document}. [ABSTRACT FROM AUTHOR]- Published
- 2024
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