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(2n - 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.
Research partially supported by NRC Twinning Program Grant R80555
Research partially supported by NRC Research Grant MDA904-95-H-1019
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
E. F. Assmus, Jr., On 2-ranks of Steiner triple systems, Electronic J. Combin., Vol. 2 (1995), paper R9.
E. F. Assmus, Jr., and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge (1992).
T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Cambridge University Press, Cambridge (1986).
M.-L. de Resmini, On k-sets of type (m, n) in a Steiner system 5(2, 1; υ), in: Finite Geometries and Designs, LMS Lecture Notes, Vol. 49 (1981), pp. 104–113.
J. Doyen, Sur la structure de certains systemes triples de Steiner, Math. Zeitschrift, Vol. 111 (1969) pp. 289–300.
J. Doyen, X. Hubaut, and M. Vandensavel, Ranks of incidence matrices of Steiner triple systems, Math. Z., Vol. 163 (1978) pp. 251–259.
H. Lenz and H. Zeitler, Arcs and ovals in Steiner triple systems, Lecture Notes in Mathematics, Vol. 969 (1982) pp. 229–250.
N. V. Semakov and V. A. Zinov’ev, Balanced codes and tactical configurations, Problemy Peredachi Informatsii, Vol. 5(3) (1969) pp. 28–36.
L. Teirlinck, On projective and affine hyperplanes, J. Combin. Theory Ser. A, Vol. 28 (1980) pp. 290–306.
V. Tonchev, Combinatorial Configurations, Longman, Wiley, New York (1988).
V. Tonchev and R. Weishaar, Steiner triple systems of order 15 and their codes, J. Stat. Plann. Inference, to appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Hanfried Lenz on the occasion of his 80th birthday
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers, Boston
About this chapter
Cite this chapter
Baartmans, A., Landjev, I., Tonchev, V.D. (1996). On the Binary Codes of Steiner Triple Systems. In: Jungnickel, D. (eds) Designs and Finite Geometries. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1395-3_3
Download citation
DOI: https://doi.org/10.1007/978-1-4613-1395-3_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4612-8604-2
Online ISBN: 978-1-4613-1395-3
eBook Packages: Springer Book Archive