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Small stopping sets in Steiner triple systems

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Abstract

The size of the smallest stopping set in a low-density parity check (LDPC) code determines, to an extent, the performance of iterative decoding methods over the binary erasure channel. In an LDPC code arising from a block design, a stopping set is a subset of its blocks with the property that every point belonging to one of the selected blocks belongs to at least two of them. While some Steiner triple systems have no stopping sets of size 5 or less, it is shown that every Steiner triple system has a stopping set of size at most 7. The proof can be viewed as an application of polynomial identities among configuration counts that generalize the family of known linear identities. While no example of a Steiner triple system whose smallest stopping set has size seven is known, it is shown that a partial Steiner triple system on v points with c·v 1.8 triples that has no stopping set of size less than 7 exists.

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Authors and Affiliations

  1. Computer Science and Engineering, Arizona State University, Tempe, AZ, 85287-8809, USA

    Charles J. Colbourn

  2. Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8601, Japan

    Yuichiro Fujiwara

Authors
  1. Charles J. Colbourn
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  2. Yuichiro Fujiwara
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Correspondence to Charles J. Colbourn.

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Colbourn, C.J., Fujiwara, Y. Small stopping sets in Steiner triple systems. Cryptogr. Commun. 1, 31–46 (2009). https://doi.org/10.1007/s12095-008-0002-y

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  • DOI: https://doi.org/10.1007/s12095-008-0002-y

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