Finite computable dimension and degrees of categoricity

We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below 0", and not above 0'. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form [0((alpha)), 0((alpha+ 1)] for any computable ordinal alpha. We then extend the technique to produce a rigid structure of computable dimension 3 such that if d(0), d(1), and d(2) are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each d(i) < d(0) circle plus d(1) circle plus d(2). The resulting structure is an example of a structure that has a degree of categoricity, but not strongly. (C) 2018 Elsevier B.V. All rights reserved.