Symmetric skew balanced starters and complete balanced Howell rotations

Abstract:

Symmetric skew balanced starters on $n$ elements have been previously constructed for $n = 4k + 3$ a prime power and $8k + 5$ a prime power. In this paper we give an approach for the general case $n = {2^m}k + 1$ a prime power with $k$ odd. In particular we show how this approach works for $m = 2$ and $3$. Furthermore, we prove that for $n$ of the general form and $k > 9 \cdot {2^{3m}}$, then a symmetric skew balanced starter always exists. It is known that a symmetric skew balanced starter on $n$ elements, $n$ odd, can be used to construct complete balanced Howell rotations (balanced Room squares) for $n$ players and $2(n + 1)$ players, and in the case that $n$ is congruent to $3$ modulo $4$, also for $n + 1$ players.