New constructions for balanced Howell rotations

Howell rotations have been used in bridge tournaments for a long time. But it was not until 1955 that Parker and Mood first gave a rigorous definition of a balanced Howell rotation and began a systematic study of its mathematical properties. Later, Berlekamp and Hwang extended this work to the study of complete balanced Howell rotations (which are special cases of balanced Howell rotations). Surprisingly, even though the concept of balanced Howell rotations precedes that of complete balanced Howell rotations, systematic construction methods have been studied only for the latter. Most of these construction methods use the properties of a Galois field GF(p^γ) where p^γ is a prime power. In this paper, we use the properties of a Galois domain GD(p^γq^s) to construct balanced Howell rotations for n partnerships where n ? 1 is the product of two prime powers satisfying certain conditions. In particular, we construct a balanced Howell rotation for 36 partnerships, this being the smallest number for which the existence of a balanced Howell rotation was not previously known. We also give two composition methods for the constructions of balanced Howell rotations.