| A permutation array is a set of permutations on n symbols. A permutation array is k-transitive, denoted by t-PA(n,k), if for any k-tuple of positions rho=(p 1,p2,...,pk) and any k-tuple of symbols tau=(t 1,t2,...,tk), there is a permutation pi in t-PA(n,k) that maps rho to tau. A code permutation array has hamming distance d, denoted by c-PA(n,d), if any two distinct permutations in this c-PA(n,d) differ in at least d positions. When there exists a sharply k-transitive group, a minimum t-PA(n,k) and a maximum c-PA(n,d) where d=n-k+1 are achieved. However, except for the trivial groups, i.e., symmetric, alternating and cyclic groups, and the Mathieu groups M11, M12 which are sharply 4-transitive and sharply 5-transitive, respectively, the nonexistence of sharply k-transitive groups was proved for all k > 3 and for infinitely many values of n when k = 2 or k = 3. We consider methods to lower the size of transitive permutation arrays in those cases. We also give new techniques to increase the size of code permutation arrays for given hamming distances. |
