THE EXISTENCE OF r x 4 GRID-BLOCK DESIGNS WITH r=3, 4

Abstract

For a v-set V, let A be a collection of r x c arrays with elements in V. A pair (V, A) is called an r x c grid-block design if every two distinct elements i and j in V occur exactly once in the same row or in the same column of an array in A. This design originated from the use of DNA library screening. In this paper, we show the existence of r x 4 grid-block designs with r = 3, 4. We settle completely for the case of r = 4 and almost completely for the case of r = 3, leaving 15 orders undetermined.