Near triple arrays

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A Pub Date : 2025-10-02 DOI:10.1016/j.jcta.2025.106121

Alexey Gordeev, Klas Markström, Lars-Daniel Öhman

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引用次数: 0

Abstract

We introduce near triple arrays as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near triple arrays form a common generalization of such well-studied classes of designs as triple arrays, (near) Youden rectangles and Latin squares.

We enumerate near triple arrays for a range of small parameter sets and show that they exist in the vast majority of the cases considered. As a byproduct, we obtain the first complete enumerations of

6 \times 10

triple arrays on 15 symbols,

7 \times 8

triple arrays on 14 symbols and

5 \times 16

triple arrays on 20 symbols.

Next, we give several constructions for families of near triple arrays, and e.g. show that near triple arrays with 3 rows and at least 6 columns exist for any number of symbols. Finally, we investigate a duality between row and column intersection sizes of a row-column design, and covering numbers for pairs of symbols by rows and columns. These duality results are used to obtain necessary conditions for the existence of near triple arrays. This duality also provides a new unified approach to earlier results on triple arrays and balanced grids.

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近三元数组

我们引入近三重数组作为二进制行-列设计，对于符号的复制数，对于行对，列对和行与列对的相交大小，具有最多两个连续值。近三组数组形成了诸如三组数组、（近）约登矩形和拉丁正方形等设计的共同概括。我们列举了一系列小参数集的近三重数组，并表明它们存在于所考虑的绝大多数情况下。作为副产品，我们获得了第一个完整的枚举：6×10 15个符号上的三元数组，7×8 14个符号上的三元数组和5×16 20个符号上的三元数组。其次，我们给出了近三列数组族的几种构造，并举例说明了任意数目的符号都存在3行至少6列的近三列数组。最后，我们研究了行-列设计的行和列相交大小之间的对偶性，以及行和列对符号的覆盖数。利用这些对偶结果，得到了近三元数组存在的必要条件。这种对偶性还提供了一种新的统一方法来处理三重阵列和平衡网格的早期结果。

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 Theory Series A 数学-数学

CiteScore

2.90

自引率

9.10%

发文量

审稿时长

12 months

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.