In this paper we consider an error-detecting code based on linear quasigroups. Namely, each input block a0a1…an-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_0a_1\ldots a_{n-1}$$\end{document} is extended into a block a0a1…an-1d0d1…dn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_0a_1\ldots a_{n-1}d_0d_1\ldots d_{n-1}$$\end{document}, where the redundant characters d0,d1,…,dn-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_0, d_1, \ldots , d_{n-1}$$\end{document} are defined with di=ai∗ai+1∗ai+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_i=a_i*a_{i+1}*a_{i+2}$$\end{document}, where ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document} is a linear quasigroup operation and the operations in the indexes are modulo n. We give a proof that under some conditions the code is linear. Using this fact, we contribute to the determination of the error-detecting capability of the code. Namely, we determine the Hamming distance of the code and from there we obtain the number of errors that the code will detect for sure when linear quasigroups of order 4 from the best class of quasigroups of order 4 for which the constant term in the linear representation is zero matrix are used for coding. All results in the paper are derived for arbitrary length of the input blocks. With the obtained results we showed that when a small linear quasigroup of order 4 from the best class of quasigroups of order 4 is used for coding, the number of errors that the code surely detects is upper bounded with 4.

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关于线性拟群码的检错能力

在本文中，我们考虑一种基于线性拟群的检错码。即，每个输入块 a0a1…an-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage {upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_0a_1\ldots a_{n-1}$$\end{document} 扩展到一个块 ​​a0a1…an-1d0d1…dn-1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{ -69pt} \begin{document}$$a_0a_1\ldots a_{n-1}d_0d_1\ldots d_{n-1}$$\end{document}，其中冗余字符d0,d1,..., 其中 ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin }{-69pt} \begin{document}$$*$$\end{document} 是一个线性拟群运算，索引中的运算是模 n。我们证明在某些条件下代码是线性的。利用这一事实，我们有助于确定代码的错误检测能力。即，我们确定代码的汉明距离，并从那里我们获得代码将确定的错误数量，当 4 阶线性拟群来自最好的 4 阶拟群类时，线性表示中的常数项是零矩阵用于编码。论文中的所有结果都是针对任意长度的输入块导出的。得到的结果表明，当使用 4 阶拟群的最佳类中的一个小的 4 阶线性拟群进行编码时，该编码确定检测到的错误数的上限为 4。