Designs and Perfect Codes
A Steiner triple system of order n (briefly STS(n)) is a family of 3-element blocks (subsets or triples) of the set N={1,2,..., n} such that each not ordered pair of elements of N appears in exactly one block. 
 
Two STS-s of order n are called isomorphic if there exists a permutation on the set N which transforms them into one another. It is well known that STS(n) exists if and only if n≡1 or 3 (mod 6).
There exist Steiner triple systems of order 15 that do not occur in a perfect binary one‐error‐correcting code
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. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 465–468, 2007
All 80 steiner triple systems on 15 elements are derived
We settle the remaining fourteen cases by exhibiting quadruple systems SQS(16) that contain the triple systems STS(15) in question and briefly describe the techniques used to construct these systems. We also state some further computational results that we obtained as a by-product.