Balanced Ouchterlony Neighbour Designs and Quasi Rees Neighbour Designs

This paper concerns neighbour designs in which the elements of each block are arranged on the circumference of a circle. Most of the designs considered comprise a general class of balanced Ouchterlony neighbour designs, which include the balanced circuit designs of Rosa and Huang \([30]\), the neighbour designs of Rees \([29]\), and the more general neighbour designs of Hwang and Lin \([13]\). The class of Rees neighbour designs includes schemes given in 1892 by Lucas \([22]\) for round dances. Isomorphism, species, and adjugate set are defined for balanced Ouchterlony neighbour designs, and some seemingly new methods of constructing such designs are presented. A new class of quasi Rees neighbour designs is defined to cover a situation where Rees neighbour designs cannot exist but where a next best thing may be needed by experimental scientists. Even-handed quasi Rees neighbour designs and even-handed balanced Ouchterlony neighbour designs are defined too, the latter being closely related to serially balanced sequences. This paper does not provide a complete survey of known results, but aims to give the flavour of the subject and to indicate many openings for further research.