Strong External Difference Families and Classification of α -Valuations

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2025-06-16 DOI:10.1002/jcd.21985

Donald L. Kreher, Maura B. Paterson, Douglas R. Stinson

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引用次数: 0

Abstract

One method of constructing $(a^{2} + 1, 2, a, 1)$ -SEDFs (i.e., strong external difference families) in $Z_{a^{2} + 1}$ makes use of $α$ -valuations of complete bipartite graphs $K_{a, a}$ . We explore this approach and we provide a classification theorem which shows that all such $α$ -valuations can be constructed recursively via a sequence of “blow-up” operations. We also enumerate all $(a^{2} + 1, 2, a, 1)$ -SEDFs in $Z_{a^{2} + 1}$ for $a \leq 14$ and we show that all these SEDFs are equivalent to $α$ -valuations via affine transformations. Whether this holds for all $a > 14$ as well is an interesting open problem. We also study SEDFs in dihedral groups, where we show that two known constructions are equivalent.

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强外部差分族与α -估值分类

一种构造2 + 1的方法，2, a, 1) - sedf(即强外差族)在za2 + 1中利用了α-完全二部图K的赋值我们探索了这种方法，并提供了一个分类定理，该定理表明所有这样的α -赋值都可以通过一系列“爆破”操作递归地构造。我们也枚举所有（a 2 + 1）2, a，1) -SEDFs在za2 + 1对于a≤14，我们证明了所有这些sedf通过仿射变换等价于α -值。这是否对所有人都适用？14也是一个有趣的开放问题。我们还研究了二面体基团中的sedf，在那里我们证明了两个已知的结构是等效的。

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.