Advertisement

Designs, Codes and Cryptography

, Volume 86, Issue 3, pp 449–462 | Cite as

On Bonisoli’s theorem and the block codes of Steiner triple systems

  • Dieter JungnickelEmail author
  • Vladimir D. Tonchev
Article

Abstract

A famous result of Bonisoli characterizes the equidistant linear codes over \({\mathrm{GF}}(q)\) (up to monomial equivalence) as replications of some q-ary simplex code, possibly with added 0-coordinates. We first prove a variation of this theorem which characterizes the replications of first order generalized Reed–Muller codes among the two-weight linear codes. In the second part of this paper, we use Bonisoli’s theorem and our variation to study the linear block codes of Steiner triple systems, which can only be non-trivial in the binary and ternary case. Assmus proved that the block by point incidence matrices of all Steiner triple systems on v points which have the same 2-rank generate equivalent binary codes and gave an explicit description of a generator matrix for such a code. We provide an alternative, considerably simpler, proof for these results by constructing parity check matrices for the binary codes spanned by the incidence matrix of a Steiner triple system instead, and we also obtain analogues for the ternary case. Moreover, we give simple alternative coding theoretical proofs for the lower bounds of Doyen, Hubaut and Vandensavel on the 2- and 3-ranks of Steiner triple systems.

Keywords

Steiner triple system Linear code Two-weight code 

Mathematics Subject Classification

05B05 51E10 94B27 

Notes

Acknowledgements

The authors wish to thank the unknown referees for reading carefully the manuscript and making several useful remarks. The authors wish to thank also Luc Teirlinck for allowing them to include his Theorem 5.9 and Harold Ward for pointing out reference [6]. This work was done while the second author was visiting the University of Augsburg as an Alexander von Humboldt Research Fellow. Vladimir Tonchev thanks the University of Augsburg for the kind hospitality, and acknowledges support by the Alexander von Humboldt Foundation and NSA Grant H98230-16-1-0011.

References

  1. 1.
    Assmus Jr. E.F.: On \(2\)-ranks of Steiner triple systems. Electron. J. Comb. 2, R9 (1995).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Assmus Jr. E.F., Key J.D.: Designs and their codes. Cambridge University Press, Cambridge (1992).CrossRefzbMATHGoogle Scholar
  3. 3.
    Assmus Jr. E.F., Mattson Jr. H.F.: Error-correcting codes: an axiomatic approach. Inform. Control 6, 315–330 (1963).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beth T., Jungnickel D., Lenz H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999).CrossRefzbMATHGoogle Scholar
  5. 5.
    Bonisoli A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Comb. 18, 181–186 (1983).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Borges J., Rifà J., Zinoviev V.A.: On \(q\)-ary linear completely regular codes with \(\rho = 2\) and antipodal dual. Adv. Math. Commun. 4, 567–578 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Calderbank R., Kantor W.M.: The geometry of two-weight codes. Bull. Lond. Math. Soc. 18, 97–122 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coupland J.: On the construction of certain Steiner systems. Ph. D. thesis, University College of Wales (1975).Google Scholar
  9. 9.
    De Clerck F., Durante N.: Constructions and characterizations of classical sets in \(PG(n,q)\). In: Current Research Topics in Galois Geometry, pp. 1–33. Nova Science Publishers, New York (2012).Google Scholar
  10. 10.
    Delsarte P.: An algebraic approach to the association schemes of coding theory. Ph. D. thesis, Philips Research Laboratories (1973).Google Scholar
  11. 11.
    Doyen J.: Sur la structure de certains systèmes triples de Steiner. Math. Z. 111, 289–300 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Doyen J., Hubaut X., Vandensavel M.: Ranks of incidence matrices of Steiner triple systems. Math. Z. 163, 251–259 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hall Jr. M.: Combinatorial Theory, 2nd edn. Wiley, New York (1986).zbMATHGoogle Scholar
  14. 14.
    Hamada N.: On the \(p\)-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes. Hiroshima Math. J. 3, 153–226 (1973).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRefzbMATHGoogle Scholar
  16. 16.
    Jungnickel D.: Recent results on designs with classical parameters. J. Geom. 101, 137–155 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jungnickel D.: Characterizing geometric designs, II. J. Comb. Theory Ser. A 118, 623–633 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jungnickel D., van Lint J.H. (Eds.): Designs and Codes—A memorial tribute to Ed Assmus. Des. Codes Cryptogr. 17 and 18 (1999).Google Scholar
  19. 19.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, New York (1977).zbMATHGoogle Scholar
  20. 20.
    Teirlinck L.: Combinatorial structures. Ph. D. Thesis, University of Brussels, Brussels (1976).Google Scholar
  21. 21.
    Teirlinck L.: On projective and affine hyperplanes. J. Comb. Theory Ser. A 28, 290–306 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tonchev V.D.: Linear perfect codes and a characterization of the classical designs. Des. Codes Cryptogr. 17, 121–128 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tonchev V.D.: A mass formula for Steiner triple systems STS\((2^{n}-1)\) of 2-rank \(2^n -n\). J. Comb. Theory Ser. A 95, 197–208 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tonchev V.D., Weishaar R.S.: Steiner triple systems of order 15 and their codes. J. Stat. Plan. Inference 58, 207–216 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    van Lint J.H.: Introduction to Coding Theory, 3rd edn. Springer, New York (1999).CrossRefzbMATHGoogle Scholar
  26. 26.
    Ward H.N.: An introduction to divisible codes. Des. Codes Cryptogr. 17, 73–79 (1999).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of AugsburgAugsburgGermany
  2. 2.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA

Personalised recommendations