Terraces for Small Groups

We use heuristic algorithms to find terraces for small groups. We show that Bailey’s Conjecture (that all groups other than the non-cyclic elementary abelian 2-groups are terraced) holds up to order 511, except possibly at orders 256 and 384. We also show that Keedwell’s Conjecture (that all non-abelian groups of order at least 10 are sequenceable) holds up to order 255, and for the groups \(A_6, S_6, PSL(2, q_1)\) and \(PGL(2, q_2)\) where \(q_1\) and \(q_2\) are prime powers with \(3 \leq q_1 \leq 11\) and \(3 \leq q_2 \leq 8\). A sequencing for a group of a given order implies the existence of a complete Latin square at that order. We show that there is a sequenceable group for each odd order up to 555 at which there is a non-abelian group. This gives 31 new orders at which complete Latin squares are now known to exist, the smallest of which is 63.

In addition, we consider terraces with some special properties, including constructing a directed \(T_2\)-terrace for the non-abelian group of order 21 and hence a Roman-2 square of order 21 (the first known such square of odd order). Finally, we report the total number of terraces and directed terraces for groups of order at most 15.