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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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