Elements of the hyperoctahedral group of
given norm and deviation

Abstract. The \(n\)-dimensional hyperoctahedral group, denoted by \(T_{n}\;\), is the group of the
distance-preserving transforms of the \(n\)-dimensional cube. In this group, a norm and a deviation of an element can be defined as the maximum and minimum of the distances of each vertex and its transformed one. In previous papers the author proved results on the computation of these values and characterized the image of the norm and deviation functions. In the present paper we deal with the following problem: which ordered pair \((p,q)\) belongs to a transform \(u\) so that the norm and the deviation of \(u\) is equal to \(p\) and \(q\), respectively.