A new infinite series of double Youden rectangles

Bailey (1989) defined a k x v double Youden rectangle (DYR), with L < v, as a type of balanced Graeco-Latin design where each Roman letter occurs exactly once in each of the k rows of the rectangle, and each Greek letter occurs exactly once in each of the v columns. A DYR of a particular size k x v can exist only if there exists a symmetric 2-design for v treatments in blocks of size k, but existence of a symmetric 2-design does not guarantee the existence of a corresponding DYR, nor does it provide a construction for such a DYR. Vowden (1994) provided constructions of DYRs of sizes k x (2k + 1) where k > 3 is a prime power with k = 3 (modulo 4). We now provide a general construction for DYRs of sizes k x (2k +1) where k > 5 is a prime power with k = 1 (modulo 4). We present DYRs of sizes 9 x 19 and 13 x 27.