A Markov chain for Steiner triple systems

These notes describe what might be a Markov chain method for choosing a random Steiner triple system. Many things are not known, including whether or not the Markov chain is connected! I include a positive result of Grannell and Griggs according to which any two isomorphic Steiner triple systems lie in the same connected component. 1 Choosing at random Suppose I have a fair coin. How can I choose a random Steiner triple system on 103 points? A fair coin is a device which can in one step (or toss) produce one bit of information (a 0 or a 1, or informally, “heads” or “tails”), in such a way that the results of different tosses are independent – this means that each of the 2n sequences of results produced by n tosses occurs with the same probability, namely 1/2n. Given a fair coin, there is a simple algorithm for choosing a random integer x in the range [0,2n−1]. We just toss the coin n times and interpret the sequence of bits as the expansion of x in base 2. What about choosing an integer in the range [0,N−1], where N is arbitrary? We cannot do this with a bounded number of coin tosses if N is not a power of 2, since the probability of any event defined by n coin tosses is a rational number with denominator 2n. So we have to make a small compromise, as follows. Choose n to be the least integer such that 2n ≥ N. Choose an integer in the range [0,2n−1] as before. If it is smaller than N, we accept this result; otherwise we try agin, and continue until a result is obtained. It is not hard to show that, if p = N/2n, then • the resulting integer is uniformly distributed in the range [0,N−1];