Skip to main content
Springer Nature Link
Log in

A complete classification of symmetric (31, 10, 3) designs

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In recent years several authors have determined all symmetric (31, 10, 3) designs with a nontrivial automorphism. Here we describe an algorithm for the generation by computer of all symmetric (31, 10, 3) designs and find that there are precisely 151 such nonisomorphic designs. Of these, 107 have a trivial automorphism group.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beth, Th., Jungnickel, D., Lenz, H. 1985. Design Theory, B.I.-Wissenschaftsverlag, Mannheim.

    Google Scholar 

  2. Colbourn, C.J., Colbourn, M.J., Harms, J.J., and Rosa, A. 1983. A complete census of (10, 3, 2)-block designs and of Mendelsohn triple systems of order 10, III. (10, 3, 2)-block designs without repeated blocks. Congr. Numer. 39:211–234.

    Google Scholar 

  3. Denniston, R.H.F. 1982. Enumeration of symmetric designs (25, 9, 3). Annals of Discrete Math., 15:111–127.

    Google Scholar 

  4. Hall, M. Jr., 1967. Combinatorial Theory, Waltham: Blaisdell.

    Google Scholar 

  5. van Lint, J.H. and Tonchev, V.D. A class of non-embeddable designs. J. Combin. Theory (A), to appear.

  6. Mathon, R.A. 1988. Symmetric (31, 10, 3) designs with nontrivial automorphism group. Ars Combinatoria 25:171–183.

    Google Scholar 

  7. Mathon, R.A. and Rosa, A. 1985. Table of parameters of BIBD's with r ≤ 41 including existence, enumeration and resolvability results. Annals of Discrete Math., 26:275–308.

    Google Scholar 

  8. Spence, E. 1991. Symmetric (31, 10, 3) designs with a nontrivial automorphism of odd order. J. Comb. Math. and Comb. Comp., 10: 51–64.

    Google Scholar 

  9. Tonchev, V.D. 1987. Symmetric (31, 10, 3) designs with an automorphism of order 7. Annals of Discrete Math., 34:461–464.

    Google Scholar 

  10. Tonchev, V.D. 1988. Symmetric designs without ovals and extremal self-dual codes. Annals of Discrete Math., 37:451–458.

    Google Scholar 

  11. Tonchev, V.D. Symmetric designs with trivial automorphism group, to appear in Ars Combinatoria.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Glasgow, G12 8QQ, Glasgow, Scotland

    Edward Spence

Authors
  1. Edward Spence
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by D. Jungnickel

Rights and permissions

Reprints and permissions

About this article

Cite this article

Spence, E. A complete classification of symmetric (31, 10, 3) designs. Des Codes Crypt 2, 127–136 (1992). https://doi.org/10.1007/BF00124892

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00124892

Keywords