Double Youden rectangles of sizes \(p \times (2p +1)\) and \((p+ 1) \times (2p + 1)\)

A \(k \times v\) double Youden rectangle (DYR) is a type of balanced Graeco-Latin design where each Roman letter occurs exactly once in each of the \(k\) rows, where each Greek letter occurs exactly once in each of the \(v\) columns, and where each Roman letter is paired exactly once with each Greek letter. The other properties of a DYR are of balance, and indeed the structure of a DYR incorporates that of a symmetric balanced incomplete block design (SBIBD). Few general methods of construction of DYRs are known, and these cover only some of the sizes \(k \times v\) with \(k = p\) (odd) or \(p+1\), and \(v = 2p + 1\). Computer searches have however produced DYRs for those such sizes, \(p \leq 11\), for which the existence of a DYR was previously in doubt. The new DYRs have cyclic structures. A consolidated table of DYRs of sizes \(p \times (2p +1)\) and \((p +1) \times (2p +1)\) is provided for \(p \leq 11\); for each of several of the sizes, DYRs are given for different inherent SBIBDs.