Equivalence and Characterizations of Linear Rank-Metric Codes Based on Invariants
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We show that the sequence of dimensions of the linear spaces, generated by a given rank-metric code together with itself under several applications of a field automorphism, and the sequence of dimensions of the intersections of itself under several applications of a field automorphism, are invariants for the whole equivalence class of the code. These invariants give rise to easily computable criteria to check if two codes are inequivalent. We derive some concrete values and bounds for these dimension sequences for some known families of rank-metric codes, namely Gabidulin and (generalized) twisted Gabidulin codes. We then derive conditions on the length of the codes with respect to the field extension degree, such that codes from different families cannot be equivalent. Furthermore, we give an exact number of equivalence classes Gabidulin codes and derive bounds on the number of equivalence classes for twisted Gabidulin codes. In the end we use the mentioned sequences to characterize Gabidulin codes in various ways.
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