Some highly symmetric authentication perpendicular arrays

A set S of permutations of k objects is -uniform, t-homogeneous if for every pair A, B of t-subsets of the ground set, there are exactly permutations in S mapping A onto B. Arithmetical conditions and symmetries are discussed. We describe the character-theoretic method which is useful if S is contained in a permutation group. A main result is the construction of a 2-uniform, 2-homogeneous set of permutations on 6 objects and of a 3-uniform, 3-homogeneous set of permutations on 9 objects. These are contained in the simple permutation groups PSL₂(5) and PSL₂(8), respectively. The result is useful in the framework of theoretical secrecy and authentication (see Stinson 1990, Bierbrauer and Tran 1991).