Designs with partial neighbour balance (Q1063988)

A nearest neighbour (NN) design may be one-dimensional, with blocks ordered either cyclically or linearly, or it may be a two-dimensional rectangular array (row and column design), regarded as linear in each direction or as being on a torus. A one-dimensional design is said to have exact m-th-level NN balance if and only if all unordered pairs of distinct treatments occur as m-th nearest neighbours equally often, say \(\lambda_ m\) times. (The authors also define partial m-th level NN balance). This paper is motivated by the observation of \textit{G. N. Wilkinson}, \textit{S. R. Eckert}, \textit{T. W. Hancock} and \textit{O. Mayo} (see the preceding entry, Zbl 0575.62069) that efficiency is enhanced for 2-replicate designs when for any pair of treatments their lists of neighbours (to a specified level) has at most one element in common. This paper gives several constructions for designs which satisfy this balance property of Wilkinson et al.