Symmetric skew balanced starters and complete balanced Howell rotations

Abstract

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