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On the Binary Codes of Steiner Triple Systems

In: Designs and Finite Geometries

Author

Listed:
  • Alphonse Baartmans

    (Michigan Technological University, Department of Mathematical Sciences)

  • Ivan Landjev

    (Michigan Technological University, Department of Mathematical Sciences
    Institute of Mathematics)

  • Vladimir D. Tonchev

    (Michigan Technological University, Department of Mathematical Sciences)

Abstract

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

Suggested Citation

  • Alphonse Baartmans & Ivan Landjev & Vladimir D. Tonchev, 1996. "On the Binary Codes of Steiner Triple Systems," Springer Books, in: Dieter Jungnickel (ed.), Designs and Finite Geometries, pages 29-43, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-1395-3_3
    DOI: 10.1007/978-1-4613-1395-3_3
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