Abstract
The binary code spanned by the rows of the point byblock incidence matrix of a Steiner triple system STS(v)is studied. A sufficient condition for such a code to containa unique equivalence class of STS(v)'s of maximalrank within the code is proved. The code of the classical Steinertriple system defined by the lines in PG(n-1,2)(n≥3), or AG(n,3) (n≥3) is shown to contain exactly v codewordsof weight r=(v-1)/2, hence the system is characterizedby its code. In addition, the code of the projective STS(2n-1)is characterized as the unique (up to equivalence) binary linearcode with the given parameters and weight distribution. In general,the number of STS(v)'s contained in the code dependson the geometry of the codewords of weight r. Itis demonstrated that the ovals and hyperovals of the definingSTS(v) play a crucial role in this geometry. Thisrelation is utilized for the construction of some infinite classesof Steiner triple systems without ovals.
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Baartmans, A., Landjev, I. & Tonchev, V.D. On the Binary Codes of Steiner Triple Systems. Designs, Codes and Cryptography 8, 29–43 (1996). https://doi.org/10.1023/A:1018068422276
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DOI: https://doi.org/10.1023/A:1018068422276