  A systematics search for the nuclei in which the observation of discrete
hyperdeformed (HD) bands may be feasible with existing detector facilities has
been performed in the Cd isotopes within the framework of cranked relativistic
mean field theory. It was found that the $^{96}$Cd nucleus is a doubly magic HD
nucleus due to large proton Z=48 and neutron N=48 HD shell gaps. The best
candidate for experimental search of discrete HD bands is $^{107}$Cd nucleus
characterized by the large energy gap between the yrast and excited HD bands,
the size of which is only 15% smaller than the one in doubly magic HD $^{96}$Cd
nucleus.
