However, in the present study, symmetry was observed in the doping positions. For example, for 𝑥 = 2, one scandium ion can be fixed at index 1, and the remaining scandium ions can be placed at one of the three sites (i.e., 2, 6, or 8) indicated in Fig. (a). Therefore, instead of using Eq. ( ), the following constraints can be used to narrow the search space, wherein only three bits are considered (i.e., 𝑞 # , 𝑞 2 , and 𝑞 1 ):