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Designs, Codes and Cryptography

, Volume 43, Issue 2–3, pp 103–114 | Cite as

On the structure of 1-designs with at most two block intersection numbers

  • John ArhinEmail author
Article

Abstract

We introduce the notion of an unrefinable decomposition of a 1-design with at most two block intersection numbers, which is a certain decomposition of the 1-designs collection of blocks into other 1-designs. We discover an infinite family of 1-designs with at most two block intersection numbers that each have a unique unrefinable decomposition, and we give a polynomial-time algorithm to compute an unrefinable decomposition for each such design from the family. Combinatorial designs from this family include: finite projective planes of order n; SOMAs, and more generally, partial linear spaces of order (s, t) on (s + 1)2 points; as well as affine designs, and more generally, strongly resolvable designs with no repeated blocks.

Keywords

Block intersection numbers SOMAs Unrefinable decompositions Ud-types Strongly resolvable designs 

AMS Classifications

05B05 05B25 

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References

  1. 1.
    Arhin J. Further SOMA Update, http://www.maths.qmul.ac.uk/~arhin/soma/somas.htmlGoogle Scholar
  2. 2.
    Arhin J (2006) On the construction of SOMAs and related partial linear spaces. Ph.D. thesis, University of LondonGoogle Scholar
  3. 3.
    Cameron PJ. British Combinatorial Conference Problem List, http://www.maths.qmul.ac.uk/~pjc/bcc/ allprobs.pdfGoogle Scholar
  4. 4.
    Colbourn CJ and Rosa A (1999). Triple systems. Oxford Sci. Pub., Oxford zbMATHGoogle Scholar
  5. 5.
    Ionin YJ and Shrikhande MS (1998). Resolvable pairwise balanced designs. J Stat Plan Infer zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ionin YJ and Shrikhande MS (2006). Combinatorics of symmetric designs. Cambridge University Press, New mathematical monographs zbMATHGoogle Scholar
  7. 7.
    Phillips NCK and Wallis WD (1996). All solutions to a tournament problem. Congressus Numerantium 114: 193–196 zbMATHMathSciNetGoogle Scholar
  8. 8.
    Research Problems section, Problem 197(1994) Discrete Mathematics 125:407–417Google Scholar
  9. 9.
    Shrikhande MS and Sane SS (1991). Quasi-Symmetric Designs. Cambridge University Press, London mathematical society lecture notes series zbMATHGoogle Scholar
  10. 10.
    Shrikhande SS, Raghavaro D (1964) Affine α-resolvable incomplete block designs. Contributions to Statistics Pergamon Press, Oxford, pp 471–480Google Scholar
  11. 11.
    Soicher LH (1999). On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares. Electron J Combin 6(R32): 15 MathSciNetGoogle Scholar
  12. 12.
    Soicher LH. SOMA Update, http://www.maths.qmul.ac.uk/~leonard/soma/Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesQueen Mary, University of LondonLondonUK

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