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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 sizes \(q \times (2q + 1)\) and \((q+1) \times (2q+1)\), where \( (2q+1) \) is a prime power congruent to 3 (modulo 4). Any member of these series is readily constructed from an initial column whose entries are specified very simply in terms of powers of a primitive root of GF\((2q + 1)\).
We now show that, for \(q \geq 9\), initial columns for FYRs of the above sizes can be specified more generally, which allows us to obtain many more FYRs, which are unlike any that have previously appeared in the literature. We present enumerations for \(q = 9\) and \(q = 11\), and we tabulate new FYRs for these values of \(q\). We also present some new FYRs for \(q = 15\).