Symmetric (15,8,4)-designs in terms of the geometry of binary simplex codes of dimension 4

作     者：Pankov, Mark Petelczyc, Krzysztof Zynel, Mariusz 

作者机构：Univ Warm & Mazury Olsztyn Fac Math & Comp Sci SLoneczna 54 PL-10710 Olsztyn Poland Univ Białystok Fac Math Ciolkowskiego 1M PL-15245 Bialystok Poland 

出 版 物：《DESIGNS CODES AND CRYPTOGRAPHY》 (Des Codes Cryptography)

年 卷 期：2025年第93卷第6期

页      面：1761-1775页

核心收录：

学科分类：07[理学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 070101[理学-基础数学] 

基　　金：The authors are grateful to Alessandro Montinaro for useful information concerning 

主　　题：Point-line geometry Collinearity graph Simplex code Symmetric design 

摘      要：Let n=2k-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2k-1$$\end{document} and m=2k-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=2{k-2}$$\end{document} for a certain k = 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 3$$\end{document}. Consider the point-line geometry of 2m-element subsets of an n-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension k. For k = 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 4$$\end{document} the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric (n, 2m, m)-designs. The main results concern the case k=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=4$$\end{document} and give a geometric interpretation of the five well-known symmetric (15, 8, 4)-designs.