Some new infinite series of Freeman-Youden rectangles

A Freeman-Youden rectangle (FYR) is a Graeco-Latin row-column design consisting of a balanced superimposition of two Youden squares. There are well known infinite series of FYRs of size q x (2q + 1) and (q + 1) x (2q + 1) where 2q + 1 is a prime power congruent to 3 (modulo 4). However, Preece and Cameron [9] additionally gave a single FYR of size 7 x 15. This isolated example is now shown to belong to one of a set of infinite series of FYRs of size q x (2q + 1) where q, but not necessarily 2q +1, is a prime power congruent to 3 (modulo 4), q > 3; there are associated series of FYRs of size (q + 1) x (2q + 1). Both the old and the new methodologies provide FYRs of sizes 9 x (2q + 1) and (q + 1) x (2q + 1) where both q and 2q + 1 are congruent to 3 (modulo 4), q > 3; we give special attention to the smallest such size, namely 11 x 23.