Abstract
A famous result of Bonisoli characterizes the equidistant linear codes over \({\mathrm{GF}}(q)\) (up to monomial equivalence) as replications of some q-ary simplex code, possibly with added 0-coordinates. We first prove a variation of this theorem which characterizes the replications of first order generalized Reed–Muller codes among the two-weight linear codes. In the second part of this paper, we use Bonisoli’s theorem and our variation to study the linear block codes of Steiner triple systems, which can only be non-trivial in the binary and ternary case. Assmus proved that the block by point incidence matrices of all Steiner triple systems on v points which have the same 2-rank generate equivalent binary codes and gave an explicit description of a generator matrix for such a code. We provide an alternative, considerably simpler, proof for these results by constructing parity check matrices for the binary codes spanned by the incidence matrix of a Steiner triple system instead, and we also obtain analogues for the ternary case. Moreover, we give simple alternative coding theoretical proofs for the lower bounds of Doyen, Hubaut and Vandensavel on the 2- and 3-ranks of Steiner triple systems.
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Notes
This is a corrected version of the statement in [6, Cor. 3.4], where the equidistribution property is required for the codewords in E instead of the extended code. As a comparison with [6, Thm. 3.3] shows (and as confirmed by looking at examples, for instance, at the replications of first order generalized Reed–Muller codes treated in Sect. 3), this is an obvious minor error.
The AWE just amounts to double counting the pairs \(({\mathbf {c}},h)\) with \({\mathbf {c}} \in C\) and h-coordinate \(c_h \ne 0\).
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Acknowledgements
The authors wish to thank the unknown referees for reading carefully the manuscript and making several useful remarks. The authors wish to thank also Luc Teirlinck for allowing them to include his Theorem 5.9 and Harold Ward for pointing out reference [6]. This work was done while the second author was visiting the University of Augsburg as an Alexander von Humboldt Research Fellow. Vladimir Tonchev thanks the University of Augsburg for the kind hospitality, and acknowledges support by the Alexander von Humboldt Foundation and NSA Grant H98230-16-1-0011.
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Communicated by J. D. Key.
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Jungnickel, D., Tonchev, V.D. On Bonisoli’s theorem and the block codes of Steiner triple systems. Des. Codes Cryptogr. 86, 449–462 (2018). https://doi.org/10.1007/s10623-017-0406-9
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DOI: https://doi.org/10.1007/s10623-017-0406-9