A family of 4-designs on 26 points

Dragan M. Acketa; Vojislav Mudrinski

Commentationes Mathematicae Universitatis Carolinae (1996)

  • Volume: 37, Issue: 4, page 843-860
  • ISSN: 0010-2628

Abstract

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Using the Kramer-Mesner method, 4 - ( 26 , 6 , λ ) designs with P S L ( 2 , 25 ) as a group of automorphisms and with λ in the set { 30 , 51 , 60 , 81 , 90 , 111 } are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called “quasi-designs”. Actions of groups P S L ( 2 , 25 ) , P G L ( 2 , 25 ) and twisted P G L ( 2 , 25 ) are being compared. It is shown that there exist 4 - ( 26 , 6 , λ ) designs with P G L ( 2 , 25 ) , respectively twisted P G L ( 2 , 25 ) as a group of automorphisms and with λ in the set { 51 , 60 , 81 , 90 , 111 } . With λ in the set { 60 , 81 } , there exist designs which possess all three considered groups as groups of automorphisms. An overview of t - ( q + 1 , k , λ ) designs with P S L ( 2 , q ) as group of automorphisms and with ( t , k ) ∈ { ( 4 , 5 ) , ( 4 , 6 ) , ( 5 , 6 ) } is included.

How to cite

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Acketa, Dragan M., and Mudrinski, Vojislav. "A family of 4-designs on 26 points." Commentationes Mathematicae Universitatis Carolinae 37.4 (1996): 843-860. <http://eudml.org/doc/247869>.

@article{Acketa1996,
abstract = {Using the Kramer-Mesner method, $4$-$(26,6,\lambda )$ designs with $PSL(2,25)$ as a group of automorphisms and with $\lambda $ in the set $\lbrace 30,51,60,81,90,111\rbrace $ are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called “quasi-designs”. Actions of groups $PSL(2,25)$, $PGL(2,25)$ and twisted $PGL(2,25)$ are being compared. It is shown that there exist $4$-$(26,6,\lambda )$ designs with $PGL(2,25)$, respectively twisted $PGL(2,25)$ as a group of automorphisms and with $\lambda $ in the set $\lbrace 51,60,81,90,111\rbrace $. With $\lambda $ in the set $\lbrace 60,81\rbrace $, there exist designs which possess all three considered groups as groups of automorphisms. An overview of $t$-$(q+1,k,\lambda )$ designs with $PSL(2,q)$ as group of automorphisms and with $(t,k) \in \lbrace (4,5), (4,6), (5,6)\rbrace $ is included.},
author = {Acketa, Dragan M., Mudrinski, Vojislav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {block designs; orbits; projective linear group; projective special linear group; twisted projective linear group; Kramer-Mesner method; block designs; orbits; projective linear group; projective special linear group; twisted projective linear group; Kramer-Mesner method},
language = {eng},
number = {4},
pages = {843-860},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A family of 4-designs on 26 points},
url = {http://eudml.org/doc/247869},
volume = {37},
year = {1996},
}

TY - JOUR
AU - Acketa, Dragan M.
AU - Mudrinski, Vojislav
TI - A family of 4-designs on 26 points
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1996
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 37
IS - 4
SP - 843
EP - 860
AB - Using the Kramer-Mesner method, $4$-$(26,6,\lambda )$ designs with $PSL(2,25)$ as a group of automorphisms and with $\lambda $ in the set $\lbrace 30,51,60,81,90,111\rbrace $ are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called “quasi-designs”. Actions of groups $PSL(2,25)$, $PGL(2,25)$ and twisted $PGL(2,25)$ are being compared. It is shown that there exist $4$-$(26,6,\lambda )$ designs with $PGL(2,25)$, respectively twisted $PGL(2,25)$ as a group of automorphisms and with $\lambda $ in the set $\lbrace 51,60,81,90,111\rbrace $. With $\lambda $ in the set $\lbrace 60,81\rbrace $, there exist designs which possess all three considered groups as groups of automorphisms. An overview of $t$-$(q+1,k,\lambda )$ designs with $PSL(2,q)$ as group of automorphisms and with $(t,k) \in \lbrace (4,5), (4,6), (5,6)\rbrace $ is included.
LA - eng
KW - block designs; orbits; projective linear group; projective special linear group; twisted projective linear group; Kramer-Mesner method; block designs; orbits; projective linear group; projective special linear group; twisted projective linear group; Kramer-Mesner method
UR - http://eudml.org/doc/247869
ER -

References

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  1. Acketa D.M., Mudrinski V., A 4 -design on 38 points, submitted. 
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  3. Acketa D.M., Mudrinski V., Two 5 -designs on 32 points, accepted for Discrete Mathematics. Zbl0873.05010
  4. Alltop W.O., An infinite class of 4 -designs, J. Comb. Th. 6 (1969), 320-322. (1969) Zbl0169.01903MR0241316
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  7. Dautović S., Acketa D.M., Mudrinski V., A graph approach to isomorphism testing of 4 - ( 48 , 5 , λ ) designs arising from P S L ( 2 , 47 ) , submitted. 
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  9. Driessen L.M.H.E., t -designs, t ≥ 3 , Tech. Report (1978), Department of Mathematics, Eindhover University of Technology, Holland. 
  10. Gorenstein D., Finite Simple Groups, An Introduction to Their Classification, Plenum Press, New York, London, 1982. Zbl0672.20010MR0698782
  11. Huppert B., Endliche Gruppen, I. Die Grundlehren der matematischen Wissenschaften, Band 134 (1967), Springer-Verlag, Berlin, Heidelberg, New York, xii + 793 pp. MR0224703
  12. Huppert B., Blackburn N., Finite Groups, III., Die Grundlehren der matematischen Wissenschaften, Band 243 (1982), Springer-Verlag, Berlin, Heidelberg, New York, p.454. Zbl0514.20002MR0662826
  13. Janko Z., Tonchev V., Private communication, . 
  14. Kramer E.S., Leavitt D.W., Magliveras S.S., Construction procedures for t -designs and the existence of new simple 6 -designs, Ann. Discrete Math. 26 (1985), 247-274. (1985) Zbl0585.05002MR0833794
  15. Kramer E.S., Mesner D.M., t -designs on hypergraphs, Discrete Math. 15 (1976), 263-296. (1976) Zbl0362.05049MR0460143
  16. Kreher D.L., Radziszowski S.P., The existence of simple 6 - ( 14 , 7 , 4 ) designs, Jour. of Combinatorial Theory, Ser. A 43 (1986), 237-243. (1986) Zbl0647.05013MR0867649
  17. Kreher D.L., Radziszowski S.P., Simple 5 - ( 28 , 6 , λ ) designs from P S L 2 ( 27 ) , Ann. Discrete Math. 34 (1987), 315-318. (1987) MR0920656

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