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An extension theorem for Steiner systems

John L. Blanchard
1995 Discrete Mathematics  
Given a Steiner system S(2,k-1;v) with v>~vo(k), there is a 3-design Sa(3, k;v+ t) such that the derived design is 2 copies of the Steiner system for any 2 sufficiently large satisfying the standard arithmetic  ...  This theorem has applications in the construction of Steiner 3-designs.  ...  There are Jimctions vo(k) and 2o(k,v) such that for any Steiner system S(2, k-1;v) with v>~vo(k) and any 2>~20(v,k) satisfying the necessary condition (3) , there is an extension of 2 copies of the  ... 
doi:10.1016/0012-365x(93)e0190-f fatcat:uhdctkr425dobgabp4zbasg5cq

On a Theorem of Ganter

1997 Combinatorics, probability & computing  
We prove a generalization of a theorem of Ganter concerning the embedding of partial Steiner systems into Steiner systems.  ...  As an application we discuss a further version of the problem of Rosenfeld on embedding graphs into strongly regular graphs.  ...  Theorem 1. Let A be a k-partial Steiner system and n an arbitrary positive integer. For some v one can find n non-isomorphic S (2, k, v) Steiner systems each of which is an extension of A.  ... 
doi:10.1017/s0963548397002952 fatcat:ftswhubhpfclxpsad7opicdk2e

Oriented Steiner quasigroups

Izabella Stuhl
2014 Journal of Algebra and its Applications  
The approach is based upon (modified) Schreier-type f-extensions for quasigroups (cf. earlier works S, NSt, NSt2) achieved through oriented Steiner triple systems.  ...  As a justification of this concept briefly discuss an application of oriented Steiner triple systems in cryptography using oriented Steiner quasigroups.  ...  Acknowledgement The author has been supported by FAPESP Grant -process No 11/51845-5, and expresses her gratitude to IMS, University of São Paulo, Brazil, for the warm hospitality.  ... 
doi:10.1142/s0219498814500728 fatcat:3v5gfxdphbhm3bfqrqkvhfs6ji

Almost all Steiner triple systems have perfect matchings [article]

Matthew Kwan
2020 arXiv   pre-print
We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems.  ...  We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class).  ...  For an ordered partial system S ∈ O m , let O * (S) ⊆ O be the set of ordered Steiner triple systems S * such that S * m = S.  ... 
arXiv:1611.02246v5 fatcat:lsge56xdczgm5mzmd4u57dkare

Moufang's theorem for non-Moufang loops

Izabella Stuhl
2015 Aequationes Mathematicae  
We introduce a class of non-Moufang loops satisfying the Moufang's theorem.  ...  Acknowledgement The author has been supported by FAPESP Grant -process No 11/51845-5, and expresses the deep gratitude to IMS, University of São Paulo, Brazil, for the warm hospitality.  ...  An oriented Steiner loop L is a loop, for which there is an oriented STS (S, T ) with the following properties (1)-(3). (1) L is a loop extension of the group of order 2 by a Steiner loop S = S ∪ e; (2  ... 
doi:10.1007/s00010-015-0342-6 fatcat:w7p56camgrbw5lq6cmfuddnpfi

The completion of finite incomplete Steiner triple systems with applications to loop theory

Christine Treash
1971 Journal of combinatorial theory. Series A  
We show that every finite incomplete Steiner triple system (totally symmetric loop) can be embedded in a finite complete triple system (totally symmetric loop).  ...  an incomplete Steiner triple system from an incomplete TS loop.  ...  Hence by induction the theorem is true. THEOREM 2. Any finite incomplete Steiner triple system is contained in a "finite" Steiner triple system. Proof.  ... 
doi:10.1016/0097-3165(71)90030-6 fatcat:xdh3c2k5hvc65d6n5xqph3i57e

An extension of Cameron's result on block schematic Steiner systems

Tsuyoshi Atsumi
1979 Journal of combinatorial theory. Series A  
It is shown that if a Steiner system s(t, k, U) (t > 3) is block schematic, then v is bounded above by a function of k.  ...  Eiichi Bannai for suggesting me the problem which is solved in this paper. I wish to thank Dr. Peter J. Cameron for his kind advice on the refining of the proof of the Theorem.  ...  (That is, 6 forms an algebra.) AiAj = c h(i, j, h) Ah, SOME PROPERTIES OF STEINER SYSTEMS In order to prove the theorem we need the following lemmas. LEMMA 1. (Mendelsohn [3] ).  ... 
doi:10.1016/0097-3165(79)90032-3 fatcat:us4sfrfsi5cxdlgyx3wicpib5u

Page 8566 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
At the end of his paper he ob- tains rather huge lower bounds for the number of non-isomorphic Steiner systems of this kind, for example, more than 3-10!°?  ...  We determine the family of subspaces of such a Steiner system and we study its properties. Moreover, we evaluate the number of such non- isomorphic Steiner systems.”  ... 

Page 1384 of Mathematical Reviews Vol. 55, Issue 5 [page]

1978 Mathematical Reviews  
An extension of Carlitz’s identity. Proc. Amer. Math. Soc. 63 (1977), no. 1, 180-184.  ...  The main problems dis- cussed come under the headings (i) existence; (ii) isomorphism; (iii) extension and completion; (iv) disjoint Steiner systems.  ... 

Some Inequalities for Steiner Systems

R. Noda
1998 European journal of combinatorics (Print)  
We give some inequalities for Steiner systems S(t, k, v) which improve the inequality v ≥ (t + l)  ...  Tokushige for informing him of the results of J. Shönheim.  ...  Also by (1.3) and (1.4) we have the inequality v ≥ (t + 1)(k − t + 1) for Steiner system S(t, k, v) [1, 3, Theorem 3A.5].  ... 
doi:10.1006/eujc.1997.0177 fatcat:kxu6ac3ljbbvreptex5oxevrvm

A construction for Steiner 3-designs

John L Blanchard
1995 Journal of combinatorial theory. Series A  
For every v satisfying necessary arithmetic conditions we construct a Steiner 3-design S(3, q + 1; v. qn+ 1) for every n sufficiently large.  ...  Starting with a Steiner 2-design S(2, q;v), this is extended to a 3-design S;~(3, q + 1; v + 1), with index 2= qd for some d, such that the derived design is 2 copies of the Steiner 2-design.  ...  DESIGN EXTENSIONS If ~ is a family of blocks from Iv, define the derived family ~d as the set of blocks and that is, the blocks of ~ that contain v with v deleted. We call ~ an extension of @d.  ... 
doi:10.1016/0097-3165(95)90015-2 fatcat:jvtox2bpnrcn3jmdnw6kx2epoa

On 2-ranks of Steiner triple systems

E. F. Assmus Jr.
1995 Electronic Journal of Combinatorics  
Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code — called the "carrier" — which depends only on the order of the system  ...  We also discuss Steiner quadruple systems and prove an analogous existence and uniqueness theorem; in this case the binary code (corresponding to the carrier in the triple system case) is the dual of the  ...  We complete that discussion in Section 7 by stating and proving our existence and uniqueness theorem for quadruple systems of de cient 2-rank, Theorem 7.1, and giving an application to resolvable Steiner  ... 
doi:10.37236/1203 fatcat:d7ti2g23djge3pbstxj3t5f6qe

Steiner Loops of Affine Type

Giovanni Falcone, Ágota Figula, Carolin Hannusch
2020 Results in Mathematics  
Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied.  ...  AbstractSteiner loops of affine type are associated to arbitrary Steiner triple systems.  ...  We will use the abbreviation STS or STS(n) for a Steiner triple system, or for a Steiner triple system on n elements, respectively.  ... 
doi:10.1007/s00025-020-01273-6 fatcat:qc5kh5wa4jgiflxz4iw3ucdm3u

Page 9 of IBM Journal of Research and Development Vol. 4, Issue 5 [page]

1960 IBM Journal of Research and Development  
Two intermediate results are of interest: 1) A Steiner triple system has for each element an involution fixing only this element, if and only if every triangle generates an S(9), a Steiner triple system  ...  Two theorems in Section 3 relate combinatorial properties of Steiner triple systems to the existence of certain IBM JOURNAL * NOVEMBER 1960 automorphisms. lirst, if for every point of a triple system S  ... 

A construction of 2-designs from Steiner systems and extendable designs

Hiroyuki Nakasora
2010 Journal of combinatorial theory. Series A  
We give a construction of a 2-(mn 2 + 1, mn, (n + 1)(mn − 1)) design starting from a Steiner system S(2, m + 1, mn 2 + 1) and an affine plane of order n.  ...  This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals.  ...  Acknowledgments The author would like to thank Masaaki Kitazume for useful advice. The author also would like to thank Naoyuki Horiguchi for his computation in Remark 3.3.  ... 
doi:10.1016/j.jcta.2009.11.003 fatcat:tcfbqickuvfjhmjqidranhbjka
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