On the binary codes of Steiner triple systems

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The binary code spanned by the rows of the point by block incidence matrix of a Steiner triple system STS(ν) is studied. A sufficient condition for such a code to contain a unique equivalence class of STS(ν)'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 exactlyν codewords of weight r = (ν 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(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 defining STS(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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© Kluwer Academic Publishers 1996. Publisher’s version of record: https://doi.org/10.1007/BF00130567

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Designs, Codes and Cryptography