Coding with Permutations

DEFI,xrrIoN. A permutation array (PA) of degree r and size v is a set of v permutations on a set -q, I -(2 = r (thought of as v orderings of the r elements), with the property that any two distinct permutations agree in at most A positions. It will be denoted by A(r, ~A; v). As an example a l,atin square, or any cyclic set of permutations, is an ,4(r, ~ 0; r). Other types of PAs have also been considered; those with the property that any two distinct permutations agree in exactly A positions, called equidistant PAs and denoted by d(r ,A; v) and those with the property that any two distinct permutations agree in at least A positions denoted b,' .q(r, ~ A; v). These two types will not be considered here. A considerable amount of information is now known about such arrays and they have received the attention of combinatorial theorists, group theorists, and engineers, among others. Because of the diversity of the background of those interested in these structures, the research results on PAs have appeared widely scattered in the literature. This has tended to hamper further progress in the area and has led to some overlap of work. The main purpose of this paper is to briefly present the most important of these results in a unified setting to assess their significance and to lay a basis for further work. In addition some new

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