Double Youden rectangles of sizes (p + 1) × (p2 + p + 1)

Abstract

A k × υ double Youden rectangle (DYR) is a type of balanced Graeco-Latin design where each of the υ Roman letters occurs exactly once in each of the k rows, and each of the k Greek letters occurs exactly once in each of the υ columns. The structure of a k × υ DYR incorporates that of a symmetric balanced incomplete block design (SBIBD) that has υ blocks each of size k. DYRs of size (p + 1) × (p² + p + 1) are, for the first time, presented for p = 4, 5 and 7 . The already known DYR for p = 3 and the new ones for p = 5 and p = 7 are all p-cyclic in structure. However, the new DYRs for p = 4 are (p + 1)-cyclic; eight species of these 5-cyclic 5 × 21 DYRs have been found, examples of which are presented.