There exist Steiner triple systems of order 15 that do not occur in a perfect binary one-error-correcting code

摘要

The codewords at distance three from a particular codeword of a perfect binary one-error-correcting code (of length 2m-1) form a Steiner triple system. It is a longstanding open problem whether every Steiner triple system of order 2m-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.