We parametrize the isometry and equivalence classes of skew constacyclic codes over a finite field by classifying the corresponding classes of their ambient rings, and present algorithms for the parametrizations. We achieve a tight classification by taking all possible Hamming-weight preserving isomorphisms between their ambient Petit rings into account. We also count the number of equivalence classes of these rings. We present many examples where isometry is a strictly stronger relation than equivalence, that is, codes that are isometric but not equivalent.

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