Two Cyclic Supplementary Difference Sets and Optimal Designs in Linear Models

Abstract

$D$-optimal exact designs in a completely randomized statistical set-up are constructed, for comparing $n>2$ qualitative factors (treatments), making $r$ observations per treatment level in the presence of $n$ (or less) quantitative or continuous factors (regression factors or covariates) of influence. Their relation with cyclic supplementary difference sets $2-(u;k_1,k_2;\lambda )$ is shown, when $n=2u\equiv 2(\mathrm{mod}4)$, $r\equiv 1(\mathrm{mod}2)$, $r\ne 1$, $r<u$ and $k_1,k_2,\lambda$ are defined by $1\le k_1\le k_2\le (u-1)/2$, $(u-2k_1{)}^2+(u-2k_2{)}^2=2(ur+u-r)$, $\lambda =k_1+k_2–(u-r)/2$. Making use of known cyclic difference sets, the existence of a multiplier and the non-periodic autocorrelation function of two sequences, such supplementary difference sets are constructed for the first time. A list of all 201 supplementary difference sets $2-(u;k_1,k_2;\lambda )$ for $n=2u<100$ is given.