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Point-missing s-resolvable t-designs: infinite series of 4-designs with constant index
Designs, Codes and Cryptography ( IF 1.6 ) Pub Date : 2023-03-30 , DOI: 10.1007/s10623-023-01206-8
Tran van Trung

The paper deals with t-designs that can be partitioned into s-designs, each missing a point of the underlying set, called point-missing s-resolvable t-designs, with emphasis on their applications in constructing t-designs. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the \(\left( {\begin{array}{c}v +1\\ k\end{array}}\right) \) k-sets chosen from a \((v+1)\)-set X into \((v+1)\) mutually disjoint s-\((v,k,\delta )\) designs, each missing a different point of X. Among others, it is shown that the existence of a point-missing \((t-1)\)-resolvable t-\((v,k,\lambda )\) design leads to the existence of a t-\((v,k+1,\lambda ')\) design. As a result, we derive various infinite series of 4-designs with constant index using overlarge sets of disjoint Steiner quadruple systems. These have parameters 4-\((3^n,5,5)\), 4-\((3^n+2,5,5)\) and 4-\((2^n+1,5,5)\), for \(n \ge 2\), and were unknown until now. We also include a recursive construction of point-missing s-resolvable t-designs and its application.



中文翻译:

点缺失 s-可解析 t-设计:具有常数索引的无限系列 4-设计

本文涉及可以划分为s设计的t设计,每个设计都缺少基础集合的一个点,称为点缺失s可解析t设计,重点是它们在构造t设计中的应用。所考虑的问题可以被视为超大集合的概括,这些集合被定义为所有\(\left( {\begin{array}{c}v +1\\ k\end{array}}\right) \) k -sets chosen from a \((v+1)\) -set X into \((v+1)\) mutually disjoint s - \((v,k,\delta )\) designs, 每个缺失X的不同点 . 其中,表明存在点缺失\((t-1)\) -可解析t - \((v,k,\lambda )\)设计导致存在t - \( (v,k+1,\lambda ')\)设计。因此,我们使用大量不相交的 Steiner 四重系统推导了具有常数索引的各种无限系列 4-设计。这些有参数 4- \((3^n,5,5)\), 4- \((3^n+2,5,5)\)和 4- \((2^n+1,5, 5)\),对于\(n \ge 2\),直到现在才为人所知。我们还包括点缺失s -可解析t -设计的递归构造及其应用。

更新日期:2023-04-01
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