Large sets of Kirkman triple systems of prime power sizes
type: Item
instance of: scholarly article
title: Large sets of Kirkman triple systems of prime power sizes
zbMATH Open document ID: 1377.05019
author: Hao Zheng
author: Junling Zhou
author: Yanxun Chang
published in: Designs, Codes and Cryptography
publication date: 2017-10-27T00:00:00Z
review text: A Steiner triple system of order \(v\), STS\((v)\), is a pair \(({\mathcal X},{\mathcal B})\), where \(\mathcal X\) is a \(v\)-set and \(\mathcal B\) is a collection of triples (blocks) such that every pair of distinct elements of \(\mathcal X\) is contained in a unique triple of \(\mathcal B\). A STS\((v)\), \(({\mathcal X},{\mathcal B}\)), is said to be resolvable if the triples in \(\mathcal B\) can be partitioned into classes, \(P_1,P_2, \dots, P_{(v-1)/2}\) such that each element of \(\mathcal X\) is contained in precisely one triple of each class. A resolvable STS\((v)\) is called a Kirkman triple system of order \(v\), KTS\((v)\). It is well known that a KTS\((v)\) exists if and only if \(v\equiv 3 \pmod{6}\) [\textit{D. K. Ray-Chaudhuri} and \textit{R. M. Wilson}, Proc. Sympos. Pure Math. 19, 187--203 (1971; Zbl 0248.05009)]. Two STS\((v)\)s on the same set \(\mathcal X\) are said to be disjoint if they have no triples in common. A partition of all triples of a \(v\)-set into pairwise disjoint STS\((v)\)s is called a large set of STS\((v)\)s, denoted by LSTS\((v)\). An LSTS\((v)\) consists of \(v-2\) pairwise disjoint STS\((v)\)s. A large set of Kirkman triple systems is denoted by LKTS\((v)\), where each STS\((v)\) is a KTS\((v)\). In [Des. Codes Cryptography 83, No. 1, 23--32 (2017; Zbl 1362.05023)], the authors constructed LKTS\((q +2)\) for any admissible prime power \(q < 400\) with 12 possible exceptions. In this paper, they extend this work and provide direct constructions for 10 of these remaining cases. Their main construction, Theorem 2.2, describes the properties required for a difference family of type \((q-1, \{2,3\},1)\) to exist that will produce an LKTS\((q+2)\). They use computer searches to construct 4 designs using Theorem 2.2. They extend work in two previous papers to provide direct constructions (again based on difference families) to take care of 6 more cases. They note that their computer searches and algorithms can also be applied to…
reviewed by: Esther R. Lamken
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 6798855
zbMATH Keywords: automorphism
zbMATH Keywords: difference family
zbMATH Keywords: large set
zbMATH Keywords: Kirkman triple system
MaRDI profile type: Publication
full work available at URL: s10623-016-0315-3
OpenAlex ID: W2560245879
cites work: Some new large sets of KTS\((v)\)
cites work: Large sets of Kirkman triple systems and related designs
cites work: Sylvester's problem of the 15 schoolgirls
cites work: Q3903014
cites work: Q4826285
cites work: A construction for large sets of disjoint Kirkman triple systems
cites work: Further results on large sets of Kirkman triple systems
cites work: On large sets of disjoint Kirkman triple systems
cites work: On large sets of Kirkman triple systems and 3-wise balanced design.
cites work: Q5661892
cites work: Some infinite families of large sets of Kirkman triple systems
cites work: Another construction for large sets of Kirkman triple systems
cites work: Transitive resolvable idempotent symmetric quasigroups and large sets of Kirkman triple systems
cites work: An improved product construction for large sets of Kirkman triple systems
cites work: Direct constructions of large sets of Kirkman triple systems
cites work: New results on large sets of Kirkman triple systems
DOI: 10.1007/S10623-016-0315-3
Recommended article: Large sets of Kirkman triple systems with order \(q^n + 2\)
Recommended article: Large sets of Kirkman triple systems of orders \(2^{2 n + 1} + 1\)
Recommended article: Large sets of resolvable Mendelsohn triple systems of prime power sizes
Recommended article: Further results on large sets of Kirkman triple systems
Recommended article: On overlarge sets of Kirkman triple systems
Recommended article: New results on large sets of Kirkman triple systems
Recommended article: Large sets of Kirkman triple systems and related designs
Recommended article: On large sets of disjoint Kirkman triple systems
Recommended article: Further results on overlarge sets of Kirkman triple systems
Recommended article: Some infinite families of large sets of Kirkman triple systems
description: scientific article; zbMATH DE number 6798855