When isometry and equivalence for skew constacyclic codes coincide

Abstract: We show that the notions of $(n,\sigma)$-isometry and $(n,\sigma)$-equivalence introduced by Ou-azzou et al coincide for most skew $(\sigma,a)$-constacyclic codes of length $n$. To prove this, we show that all Hamming-weight-preserving homomorphisms between their ambient algebras must have degree one when those algebras are nonassociative. We work in the general setting of commutative base rings $S$. As a consequence, we propose new definitions of equivalence and isometry of skew constacyclic codes that exactly capture all Hamming-preserving isomorphisms, and lead to tighter classifications. In the process we determine homomorphisms between nonassociative Petit algebras, prioritizing the algebras $S[t;\sigma]/St;\sigma$, which give rise to skew constacyclic codes.