Complete balanced Howell rotations for 16 k + 12 partnerships

Abstract

A complete balanced Howell rotation for 4 n partnerships is an arrangement of 4 n elements in a square array (also known as a balanced Room square) of side 4 n − 1 such that: 1. (i) Each of the (4 n − 1) 2 cells is either empty or contains an ordered pair of distinct elements. 2. (ii) Each of the 4 n elements appears precisely once in each row and each column. 3. (iii) Each unordered pair of distinct elements occurs is exactly one cell of the array. 4. (iv) Each pair of distinct elements appears together in a block exactly 2 n − 1) times. In this paper we show that such a rotation exists for 16 k + 12 partnerships if 8 k + 5 is a prime power. Our method is to construct skew balanced starters using cyclotomy theory.