Abstract:

It is rather easy to see that the set of words at distance three from any code word in a perfect binary code of length n = 2^m - 1 defines a Steiner triple system STS(n). A long standing conjecture was that every such Steiner triple system appears from perfect codes in this way. Very recently Patric Östergård and Olli Pottonen disproved this conjecture by showing that there are Steiner triple systems STS(15) that do not appear from perfect codes. I will discuss their work as well as other results on Steiner systems and perfect codes.