The Pace code, the Mathieu group M12 and the small Witt design S(5,6,12)

A ternary [66,10,36]3-code admitting the Mathieu group M12 as a group of automorphisms has recently been constructed by N. Pace, see Pace (2014). We give a construction of the Pace code in terms of M12 as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5,6,12). We also present a proof that the Pace code does indeed have minimum distance 36.