Mixed Steiner Triple Systems With Shortest Length

IF 0.8 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2026-10-12 Epub Date: 2026-08-10 DOI:10.1002/jcd.70030

Tuvi Etzion

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引用次数: 0

Abstract

A mixed Steiner triple system is a 3-GDD which is viewed as a code with minimum Hamming distance 3. These codes are the minimum weight codewords of a 1-perfect code over a mixed alphabet, when the related codes exist, and provide the connection between 3-GDDs and coding theory. We prove that a 3-GDD of type $1^{n} k^{1} ℓ^{1}$ , where $n = k \cdot ℓ$ , with minimum distance 3 exists for every $k$ and $ℓ$ such that $n = k \cdot ℓ$ , $k \equiv 1$ or $3 (mod 6)$ , $ℓ \equiv 1$ or $3 (mod 6)$ , and $k \geq ℓ > 1$ . These designs are of the shortest possible length (smallest number of elements) for given $k$ and $ℓ$ . Other constructions for such triple systems are also presented.

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具有最短长度的混合斯坦纳三系

混合斯坦纳三重系统是一个3- gdd，它被看作是一个最小汉明距离为3的码。当相关码存在时，这些码是混合字母表上1-perfect码的最小权重码字，并提供了3- gdd与编码理论之间的联系。我们证明了一个1型的3-GDD n k 11，其中n = k·r，最小距离3存在于每一个k和r，使得n= k·r，K≡1或3 (模6)；≡1或3 (模6)；且k≥1。对于给定的k和r，这些设计具有最短的可能长度（最小的元素数量）。对这类三重系统的其他构造也作了介绍。

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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.