Optimal partitions for triples (Q1185896)

The best known method to obtain constant weight codes with distance 4 is the partitioning method on sets of \(n\)-tuples. In this paper the author considers the case \(w=3\). For \(n\equiv 0\), \(1, 2, 3\pmod 6\) the optimal partition is derived from disjoint Steiner triple systems. Optimal partitions are obtained for all orders \(n=3k+1\), \(k\equiv 1,5\pmod 6\). For \(n=3k+2\), \(k\equiv 1,5\pmod 6\) or \(k\equiv 3\pmod{12}\) a construction is presented with maximal number \(3k\), of disjoint optimal codes. Also given are construtions of partitions of order \(n=qk+i\). It is proved that for \(n=9q+2\) there exists a partition with \(9q+1\) codes.