Some New Infinite Series of Freeman-Youden Rectangles

Abstract

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 $q\times (2q+1)$ and $(q+1)\times (2q+1)$ where $2q+1$ is a prime power congruent to $3$ (modulo $4$). However, Preece and Cameron [9] additionally gave a single FYR of size $7\times 15$. This isolated example is now shown to belong to one of a set of infinite series of FYRs of size $q\times (2q+1)$ 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 $(q+1)\times (2q+1)$. Both the old and the new methodologies provide FYRs of sizes $q\times (2q+1)$ and $(q+1)\times (2q+1)$ where both $q$ and $2q+1$ are congruent to $3$ (modulo $4$), $q>3$; we give special attention to the smallest such size, namely $11\times 23$.