Cyclotomic conditions leading to new Steiner 2-designs
type: Item
instance of: scholarly article
title: Cyclotomic conditions leading to new Steiner 2-designs
zbMATH Open document ID: 0906.05007
author: Marco Buratti
published in: Finite Fields and their Applications
publication date: 1999-02-01T00:00:00Z
review text: Sometimes the validity of elementary cyclotomic conditions in a finite field is sufficient to guarantee the existence of some Steiner 2-designs with a nice group of automorphisms. For instance, R. C. Bose proved that if \(q= 12t+ 1\) (resp. \(q= 20t+1\)) is a prime power such that \(-3\) (resp. 5) is not a fourth power in \(\text{GF}(q)\), then there exists a point-regular \(S(2,4,q)\) (resp. \(S(2,5,q)\)). Cyclotomic conditions weaker than those of R. C. Bose may be found in [\textit{M. Buratti}, J. Comb. Des. 3, No. 1, 15-24 (1995; Zbl 0833.05009)]. In the present paper, the author obtained new cyclotomic conditions starting from a paper by \textit{S. Bagchi} and \textit{B. Bagchi} [J. Comb. Theory, Ser. A 52, No. 1, 51-61 (1989; Zbl 0679.05008)], where they give constructions for Steiner 2-designs generated by a relative difference family \(\mathbb{D}\) over the additive group of a Galois ring. He pointed out that the construction of S. Bagchi and B. Bagchi were previously considered by \textit{R. Mathon} [Ann. Discrete Math. 34, 353-362 (1987; Zbl 0647.05015), Theorem 3], but, unlike them, he did not give explicit cyclotomic conditions which are a consequence of the constructions. The author also pointed out that similar constructions for relative difference families were recently given by M. Greig. The method of M. Greig is very successful but needs the aid of a computer and yields difference families that have multipliers of orders smaller than those of difference families given here. The results below are the main consequences of the constructions given by the author. (i) Let \(q= 6t+1\) be a prime power and let \(3^e\) be the largest power of 3 dividing \(t\). Then, if 3 is a \(3^e\)th power but not a \(3^{e+1}\)th power in \(\text{GF}(q)\), there exists a point-regular \(S(2,7,7q)\). (ii) Let \(q= 8t+1\) be a prime power and let \(2^e\) be the largest power of 2 dividing \(t\). Then, if 2 is a \(2^{e+2}\)th power while \(1+\sqrt 2\) is a \(2^e\)th power but…
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 11T22
zbMATH DE Number: 1199964
zbMATH Keywords: cyclotomic conditions
zbMATH Keywords: Steiner 2-designs
zbMATH Keywords: group of automorphisms
zbMATH Keywords: relative difference family
reviewed by: Ratnakaram Nava Mohan
MaRDI profile type: Publication
full work available at URL: ffta.1997.0187
OpenAlex ID: W2009572298
cites work: Designs from pairs of finite fields. I: A cyclic unital U(6) and other regular Steiner 2-designs
cites work: ON THE CONSTRUCTION OF BALANCED INCOMPLETE BLOCK DESIGNS
cites work: Improving two theorems of bose on difference families
cites work: A packing problem its application to Bose's families
cites work: Q4222135
cites work: On Cyclic Steiner 2-Designs
cites work: Splittings of Abelian groups by integers
cites work: Q4255992
cites work: A survey of partial difference sets
cites work: Q3791163
cites work: Note on a question by S. Bagchi and B. Bagchi
DOI: 10.1006/FFTA.1997.0187
description: scientific article; zbMATH DE number 1199964