Summary: We study the index sets of the class of \(d\)-decidable structures and of the class of \(d\)-decidable countably categorical structures, where \(d\) is an arbitrary arithmetical Turing degree. It is proved that the first of them is \(m\)-complete \(\Sigma_3^{0,d}\), and the second is \(m\)-complete \(\Sigma_3^{0,d}\setminus\Sigma_3^{0,d}\) in the universal computable numbering of computable structures for the language with one binary predicate.