Orderings of finite fields and balanced tournaments (Q2713602)

Consider a round robin tournament, where team \(j\) plays team \(O_r(j)\) in round \(r\), \(0 \leq j \leq n-1\) and \(1 \leq r \leq n-1\). Call the tournament balanced at level \(l\), \(1 \leq l \leq n-2\), if for two teams \(i\) and \(j\) there exists a round \(r\) with \(i = O_{r-l}(O_r(j))\) (indices are computed modulo \(n-1\)). Call the tournament completely balanced, if it is balanced at all levels. NEWLINENEWLINENEWLINEFurthermore, consider an ordering \(0 = \alpha _0,\dots ,\alpha _{q-1}\) of \(\text{GF}(q)\) such that the partial sums \(\alpha _1\), \(\alpha _1 + \alpha _2\), \(\dots \), \(\alpha _1+\dots +\alpha _{q-1}\) are distinct. If \(n = q = 2^m\), then from such an ordering one can easily derive a round robin tournament balanced at level 1. This construction has been described by \textit{K. G. Russell} [Balancing carry-over effects in round robin tournaments, Biometrika 67, No.~1, 127-131 (1980)]. The paper under review describes a general method of ordering \(\text{GF}(q)\), which, for \(q = 2^m\), yields, by the Russell construction, a completely balanced tournament.