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On the binary codes of Steiner triple systems

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Abstract

The binary code spanned by the rows of the point by block incidence matrix of a Steiner triple systemSTS(ν) is studied. A sufficient condition for such a code to contain a unique equivalence class ofSTS(ν)'s of maximal rank within the code is proved. The code of the classical Steiner triple system defined by the lines inPG(n-1, 2)(n (≥) 3, orAG(n, 3(n ≥) 3 is shown to contain exactlyν codewords of weightr = (ν 1)/2, hence the system is characterized by its code. In addition, the code of the projectiveSTS(2n-1) is characterized as the unique (up to equivalence) binary linear code with the given parameters and weight distribution. In general, the number ofSTS(ν)'s contained in the code depends on the geometry of the codewords of weightr. It is demonstrated that the ovals and hyperovals of the definingSTS(ν) play a crucial role in this geometry. This relation is utilized for the construction of some infinite classes of Steiner triple systems without ovals.

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References

  1. E. F. Assmus Jr., On 2-ranks of Steiner triple systems,Electronic J. Combin., Vol. 2 (1995), paper R9.

    Google Scholar 

  2. E. F. Assmus, Jr., and J. D. Key,Designs and Their Codes, Cambridge University Press, Cambridge (1992).

    Google Scholar 

  3. T. Beth, D. Jungnickel, and H. Lenz,Design Theory, Cambridge University Press, Cambridge (1986).

    Google Scholar 

  4. M.-L. de Resmini, Onk-sets of type(m, n) in a Steiner systemS(2, 1; ν), in:Finite Geometries and Designs, LMS Lecture Notes, Vol. 49 (1981), pp. 104–113.

    Google Scholar 

  5. J. Doyen, Sur la structure de certains systemes triples de Steiner,Math. Zeitschrift, Vol. 111 (1969) pp. 289–300.

    Google Scholar 

  6. J. Doyen, X. Hubaut, and M. Vandensavel, Ranks of incidence matrices of Steiner triple systems,Math. Z., Vol. 163 (1978) pp. 251–259.

    Google Scholar 

  7. H. Lenz and H. Zeitler, Arcs and ovals in Steiner triple systems,Lecture Notes in Mathematics, Vol. 969 (1982) pp. 229–250.

    Google Scholar 

  8. N. V. Semakov and V. A. Zinov'ev, Balanced codes and tactical configurations,Problemy Peredachi Informatsii, Vol. 5(3) (1969) pp. 28–36.

    Google Scholar 

  9. L. Teirlinck, On projective and affine hyperplanes,J. Combin. Theory Ser. A, Vol. 28 (1980) pp. 290–306.

    Google Scholar 

  10. V. Tonchev,Combinatorial Configurations, Longman, Wiley, New York (1988).

    Google Scholar 

  11. V. Tonchev and R. Weishaar, Steiner triple systems of order 15 and their codes,J. Stat. Plann. Inference, to appear.

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Michigan Technological University, 49931, Houghton, Michigan, USA

    Alphonse Baartmans

  2. Department of Mathematical Sciences, Michigan Technological University, 49931, Houghton, Michigan, USA

    Ivan Landjev

  3. Department of Mathematical Sciences, Michigan Technological University, 49931, Houghton, Michigan, USA

    Vladimir D. Tonchev

Authors
  1. Alphonse Baartmans
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  2. Ivan Landjev
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  3. Vladimir D. Tonchev
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Additional information

Communicated by: D. Jungnickel

Dedicated to Hanfried Lenz on the occasion of his 80th birthday

Research partially supported by NRC Twinning Program Grant R80555

On leave from Institute of Mathematics, 1113 Sofia, Bulgaria

Research partially supported by NSA Research Grant MDA904-95-H-1019

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Baartmans, A., Landjev, I. & Tonchev, V.D. On the binary codes of Steiner triple systems. Des Codes Crypt 8, 29–43 (1996). https://doi.org/10.1007/BF00130567

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  • DOI: https://doi.org/10.1007/BF00130567

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