On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays

The purpose of this paper is twofold. A doubly resolvable Kirkman system is a (v, k, 1)-BIBD whose blocks can be resolved into two resolutions R and R' such that any resolution class from R has at most one block in common with any resolution class from R'. Room squares are examples of such systems in the case k = 2. It was not known whether such systems existed for k >= 3. In this paper, we construct infinite classes of such designs. In particular, we display several doubly resolvable Kirkman systems with v = 27 and k = 3. An equidistant permutation array (EPA) is a v x r array defined on an r-set V of symbols such that each row is a permutation of V and any two distinct rows have Hamming distance r - 1. Such arrays are closely related to the problem described above and have attracted much interest recently. A central problem of EPA's is to determine the maximum value of v for a fixed value of r. Until now, there has been only one value of r for which a direct construction existed to produce an array having v=2r+1. The present paper provides a direct construction for these arrays which have v the order of r^3^2. Hence, this establishes that the maximum value of v grows non-linearly with r. In the case of r = 13, the largest value of v previously known was 27. We display such an array with v = 36.