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Properties of the Steiner Triple Systems of Order 19

Charles J. Colbourn, Anthony D. Forbes, Mike J. Grannell, Terry S. Griggs, Petteri Kaski, Patric R. J. Östergård, David A. Pike, Olli Pottonen
2010 Electronic Journal of Combinatorics  
Properties of the 11$\,$084$\,$874$\,$829 Steiner triple systems of order 19 are examined.  ...  In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number  ...  Kiviluoto, A catalogue of the Steiner triple systems of order 19, Bull. Inst. Combin.  ... 
doi:10.37236/370 fatcat:i6c74kvi5vgoreaxmlzouxgy6y

The Steiner quadruple systems of order 16

Petteri Kaski, Patric R.J. Östergård, Olli Pottonen
2006 Journal of combinatorial theory. Series A  
Properties of the designs-including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15-are tabulated.  ...  The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1,054,163.  ...  Acknowledgments The authors are grateful to Juhani Markula and Jukka Seppänen at the Computing Centre of Helsinki University of Technology for providing computing resources.  ... 
doi:10.1016/j.jcta.2006.03.017 fatcat:eiyusuw35ffx7ezthuw7k5jhly

Steiner Triple Systems with High Chromatic Index

Darryn Bryant, Charles J. Colbourn, Daniel Horsley, Ian M. Wanless
2017 SIAM Journal on Discrete Mathematics  
It is conjectured that every Steiner triple system of order v ≠ 7 has chromatic index at most (v+3)/2 when v ≡ 3 6 and at most (v+5)/2 when v ≡ 1 6.  ...  Herein, we construct a Steiner triple system of order v with chromatic index at least (v+3)/2 for each integer v ≡ 3 6 such that v ≥ 15, with four possible exceptions.  ...  Herke for writing the code that generates random Steiner triple systems.  ... 
doi:10.1137/17m1114338 fatcat:lcvt2pl7xbgmhbboiwjbtcbk2m

Page 23 of Mathematical Reviews Vol. , Issue 82a [page]

1982 Mathematical Reviews  
The authors prove that any partial Steiner triple system of order n can be embedded in a Steiner triple system of order v, for any v=1 or 3 (mod6) with v > 4n.  ...  The authors prove that if there exists a partial Steiner triple system of order n con- taining y triples, then there exists an equitable partial Steiner triple system of order n containing y triples.  ... 

Embedding Steiner triple systems in hexagon triple systems

C.C. Lindner, Gaetano Quattrocchi, C.A. Rodger
2009 Discrete Mathematics  
The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.  ...  A hexagon triple system of order n is a pair (X, H ) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K n with vertex set X .  ...  If a Steiner triple system of order n can be embedded in a hexagon triple system of order v, then every Steiner triple system of order n can be embedded in a hexagon triple system of order v.  ... 
doi:10.1016/j.disc.2007.12.040 fatcat:vux43kr6aba47lerz4l35fylei

Page 1808 of Mathematical Reviews Vol. , Issue 85e [page]

1985 Mathematical Reviews  
known about Steiner triple systems of small orders (up to 15): listing of the systems themselves, together with the various properties they possess.  ...  No one who does research on Steiner triple systems (or more generally block designs) should be without this paper. It is the dictionary of Steiner triple systems of small orders.  ... 

A second infinite family of Steiner triple systems without almost parallel classes

Darryn Bryant, Daniel Horsley
2013 Journal of combinatorial theory. Series A  
The only previously known examples of Steiner triple systems of order congruent to 1 (mod 6) without almost parallel classes were the projective triple systems of order 2 n − 1 with n odd, and 2 of the  ...  11, 084, 874, 829 Steiner triple systems of order 19. called blocks, such that each (unordered) pair of points occurs in exactly λ blocks in B.  ...  Acknowledgements: The authors acknowledge the support of the Australian Research Council via grants DE120100040, DP120100790 and DP120103067.  ... 
doi:10.1016/j.jcta.2013.07.002 fatcat:uf2wjgnukrgkrah4lmip6gqebm

Page 5427 of Mathematical Reviews Vol. , Issue 87j [page]

1987 Mathematical Reviews  
R. (4-HWAT) 87j:05039 Steiner triple systems of order 19 associated with a certain type of projective plane of order 10. Period. Math. Hungar. 17 (1986), no. 3, 177-184.  ...  From this he cleverly constructs a Steiner triple system (STS) on 19 points with three rather restrictive properties. The author has found no such STS among the many examples he has checked.  ... 

Steiner Triple Systems and Existentially Closed Graphs

A. D. Forbes, M. J. Grannell, T. S. Griggs
2005 Electronic Journal of Combinatorics  
We investigate the conditions under which a Steiner triple system can have a 2- or 3-existentially closed block intersection graph.  ...  Acknowledgement The authors would like to thank Petteri Kaski and PatricÖstergård, Helsinki University of Technology, for making available to us listings of Steiner triple systems of order 19.  ...  This characterization implies that such graphs can exist only for a limited range of orders of Steiner triple system, possibly even just one order; 19 points.  ... 
doi:10.37236/1939 fatcat:fiw7mrm74rc67l7tbzfqv2p7ra

On 2-ranks of Steiner triple systems

E. F. Assmus Jr.
1995 Electronic Journal of Combinatorics  
$-rank from Steiner triple systems of a specified smaller order.  ...  When the Steiner triple system is of 2-rank less than the number of points of the system, the carrier organizes all the information necessary to construct directly all systems of the given order and $2  ...  In this case of Steiner triple systems of order 8 (on 19 points) the 2-rank is either 18 or 19 (from Theorem 3.4) and Brendan McKay estimates that the number of Steiner triple systems is 11 or 12 billion  ... 
doi:10.37236/1203 fatcat:d7ti2g23djge3pbstxj3t5f6qe

Countable homogeneous Steiner triple systems avoiding specified subsystems [article]

Daniel Horsley, Bridget S. Webb
2020 arXiv   pre-print
In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fraïssé limits of classes of finite Steiner triple systems avoiding certain  ...  subsystems of the original partial system.  ...  Webb wishes to acknowledge the support of a 2014 Ethel Raybould Visiting Fellowship from the School of Physical Sciences, The University of Queensland.  ... 
arXiv:2006.04605v1 fatcat:wtyf7rs3uva53pqftdta6boaiu

Optimal and Pessimal Orderings of Steiner Triple Systems in Disk Arrays [chapter]

Myra B. Cohen, Charles J. Colbourn
2000 Lecture Notes in Computer Science  
Steiner triple systems are well studied combinatorial designs that have been shown to possess properties desirable for the construction of multiple erasure codes in RAID architectures.  ...  The ordering of the columns in the parity check matrices of these codes a ects system performance.  ...  Acknowledgements Research of the authors is supported by the Army Research O ce (USA) under Grant no. DAAG55-98-1-0272 (Colbourn).  ... 
doi:10.1007/10719839_10 fatcat:6e4j6whnynfyzc7fluelvxzawy

Optimal and pessimal orderings of Steiner triple systems in disk arrays

Myra B. Cohen, Charles J. Colbourn
2003 Theoretical Computer Science  
Steiner triple systems are well studied combinatorial designs that have been shown to possess properties desirable for the construction of multiple erasure codes in RAID architectures.  ...  The ordering of the columns in the parity check matrices of these codes a ects system performance.  ...  Acknowledgements Research of the authors is supported by the Army Research O ce (USA) under Grant no. DAAG55-98-1-0272 (Colbourn).  ... 
doi:10.1016/s0304-3975(02)00634-5 fatcat:5c4dir3gqbhhdd2aii6n2ftfbi

The centre of a sloop

Diane M. Donovan
1990 The Australasian Journal of Combinatorics  
The 3-subsets are called the blocks of S. The order of the Steiner triple system is v, the size of the set S.  ...  A Steiner triple system n is said t.o be a subsystem of S if, R ~ S and the blocks of n are also blocks of S.  ...  The software mentioned in this paper was written by Martin Sharry. I wish to acknowledge the support of a Special Projects Grant from the University of Queensland. -'EFERENCES.  ... 
dblp:journals/ajc/Donovan90 fatcat:hmwwhoppdjg2djyzyi5q5vp2vq

Page 8483 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
Summary: “The automorphism group of the Steiner triple system of order v = 6s +3, obtained from the Bose construction using any abelian group G of order 2s +1, is determined.  ...  [Rosa, Alexander] (3-MMAS-MS; Hamilton, ON) Specialized block-colourings of Steiner triple systems and the upper chromatic index. (English summary) Graphs Combin. 19 (2003), no. 3, 335-345.  ... 
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