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There exist Steiner triple systems of order 15 that do not occur in a perfect binary one‐error‐correcting code
Author(s) -
Östergård Patric R. J.,
Pottonen Olli
Publication year - 2007
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20122
Subject(s) - steiner system , mathematics , combinatorics , counterexample , order (exchange) , binary number , code (set theory) , code word , discrete mathematics , arithmetic , algorithm , computer science , decoding methods , set (abstract data type) , finance , economics , programming language
The codewords at distance three from a particular codeword of a perfect binary one‐error‐correcting code (of length 2 m −1) form a Steiner triple system. It is a longstanding open problem whether every Steiner triple system of order 2 m −1 occurs in a perfect code. It turns out that this is not the case; relying on a classification of the Steiner quadruple systems of order 16 it is shown that the unique anti‐Pasch Steiner triple system of order 15 provides a counterexample. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 465–468, 2007