Linear and circular single-change covering designs revisited

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2023-06-01 DOI:10.1002/jcd.21885

Amanda Chafee, Brett Stevens

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引用次数: 0

Abstract

A single-change covering design (SCCD) is a $v$ -set $X$ and an ordered list $ℒ$ of $b$ blocks of size $k$ where every pair from $X$ must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single-change covering design, or more simply, a single-change covering design. A single-change covering design is circular when the first and last blocks also differ by one element. A single-change covering design is minimum if no other smaller design can be constructed for a given $v, k$ . In this paper, we use a new recursive construction to solve the existence of circular SCCD( $v, 4, b$ ) for all $v$ and three residue classes of circular SCCD( $v, 5, b$ ) modulo 16. We solve the existence of three residue classes of SCCD $(v, 5, b)$ modulo 16. We prove the existence of circular SCCD $(2 c (k - 1) + 1, k, c^{2} (2 k - 2) + c)$ , for all $c \geq 1, k \geq 2$ , using difference methods.

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重新审视线性和圆形单次变更覆盖设计

单个变更覆盖设计（SCCD）是一个v$v$集X$X$和一个有序列表ℒ $k$k$大小的b$b$块的{\rm{\mathcalL}}$，其中X$X$中的每一对必须出现在至少一个块中。每对连续块的不同之处仅在于一个元素。这是一种线性的单一变更覆盖设计，或者更简单地说，是一种单一变更覆盖的设计。当第一块和最后一块也因一个元素而不同时，单个变更覆盖设计是圆形的。如果不能为给定的v，k$v，k$构造其他较小的设计，则单个变更覆盖设计是最小的。在本文中，我们使用一个新的递归构造来求解所有v$v$的循环SCCD（v，4，b$v，4、b$）的存在性以及循环SCCD的三个残基类（v，5，b$v，5、b$）模16。我们求解了SCCD（v，5，b）的三个残基类的存在性$（v，5，b）$模16。我们证明了循环SCCD（2c）的存在性（k−1）+1，c 2（2 k−2）+c）$（2c（k-1）+1，k，｛c｝^｛2｝（2k-2）+c）$，对于所有c≥1，k≥2$c\ge 1，k\ge 2$，使用差分法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.