  In this paper, we start by giving the definitions and basic facts about
hyperoctahedral number system. There is a natural correspondence between the
integers expressed in the latter and the elements of the hyperoctahedral group
when we use the inversion statistic on this group to code the signed
permutations. We use this correspondence to define a way with which the signed
permutations group can be ordered. With this classification scheme, we can find
the r-th signed permutation from a given number r and vice versa without
consulting the list in lexicographical order of the elements of the signed
permutations group.
