Non existence of optimal covariate matrices for a symmetric BIBD with non Youden layout in the presence of neighbor effects

We consider a non Youden-type layout for one solution of a symmetric BIBD (7, 7, 4, 4, 2). We introduce a block design with a circular model, following Kunert (1984), involving the left- and right-neighbor effects. We also introduce quantitative covariates assuming values in [-1, 1]. Whereas for any Youden-type layout, it is not difficult to argue that there are three orthogonal and optimal covariates’ allocations involving values (-1) and (1), for the assumed non Youden-type layout, we attempt to prove non existence of even a single optimal covariate allocation involving values (-1) and (1). This demonstrates that in the presence of left- and right-NEs, (orthogonal) determination of optimal covariates’ allocations is impossible for certain combinatorial arrangements of SBIBDs–admitting non Youden-type layouts.