A Construction for Equidistant Permutation Arrays of Index One

Abstract

An equidistant permutation array (EPA) is a v x I array in which every row is a permutation of the integers 1,2,..., r and every pair of distinct rows has precisely h columns in common. We denote such an array by A(r, h; v) and call X the index of the array. Bolton [2] defines R(r, X) to be the maximum v for which there exists an A(r, X; v). It has been shown by Deza [4] that R(r, A) < max{h + 2, k2 + k + I} where k = r h. In [5] it is shown that R(r, 1) < r(r 3) and in [6] that for any integer r > 2, R(2r + 1, 1) 3 2r + 2. In [7], construction techniques for A(r, 2; r)‘s, r a prime power, are described. This paper gives a construction for EPAs which implies R(n2 + n + 2, 1) 2 2n2 + n for n a prime power. A generalized Room square (GRS) is an r x r array defined on a symbol set V of cardinality v such that every element of V is contained in every row and column of the array precisely once, every cell contains a subset (possibly empty) of V and every pair of distinct elements of V is contained in X of the cells. Such an array will be denoted S(r, A; v). An (r, X) design is a collection B of subsets (called blocks) taken from a finite set V of elements (called varieties) such that every variety is contained in precisely r blocks and every distinct pair of varieties is contained in exactly X of the blocks. If the cardinality of V is v we denote an (r, h) design by D(r, h; v). A D(r, X; U) is called a resolvable (r, h) design (RD(r, A; v)) if the blocks of the design can be partitioned into classes (called resolution classes)

DOI: 10.1016/0097-3165(77)90039-5

Cite this paper

@article{Vanstone1977ACF, title={A Construction for Equidistant Permutation Arrays of Index One}, author={Scott A. Vanstone and Paul J. Schellenberg}, journal={J. Comb. Theory, Ser. A}, year={1977}, volume={23}, pages={180-186} }