Some series of cyclic balanced hyper-graeco-Latin superimpositions of three Youden squares Original Research Article

General constructions are provided for some cyclic balanced hyper-Graeco-Latin superimpositions of three Youden squares. These superimpositions are all row column designs of sizes q × (2q + 1) and (q + 1) × (2q + 1) where 2q + 1 is a prime power congruent to 3 (modulo 4). For 2q + 1 ⩾ 11, the designs of each size fall into three combinatorially (and statistically) distinct classes. The basic constructions, which extend constructions by Potthoff (1963) and Agrawal and Sharma (1978), involve systematic use of successive even powers and successive odd powers of a primitive element of GF(2q + 1). However, we illustrate how an idea taken from Preece and Phillips (1997) can be extended to produce some slightly more involved variants of the constructions when q is composite and sufficiently large.