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We show that the sequence of dimensions of the linear spaces, generated by agiven rank-metric code together with itself under several applications of afield automorphism, is an invariant for the whole equivalence class of thecode. The same property is proven for the sequence of dimensions of theintersections of itself under several applications of a field automorphism.These invariants give rise to easily computable criteria to check if two codesare inequivalent. We derive some concrete values and bounds for these dimensionsequences for some known families of rank-metric codes, namely Gabidulin and(generalized) twisted Gabidulin codes. We then derive conditions on the lengthof the codes with respect to the field extension degree, such that codes fromdifferent families cannot be equivalent. Furthermore, we derive upper and lowerbounds on the number of equivalence classes of Gabidulin codes and twistedGabidulin codes, improving a result of Schmidt and Zhou for a wider range ofparameters. In the end we use the aforementioned sequences to determine acharacterization result for Gabidulin codes.

Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants