When isometry and equivalence for skew constacyclic codes coincide
Monica Nevins, Susanne Pumpluen
TL;DR
The paper analyzes when $(\mathbf{n},\sigma)$-isometry and $(\mathbf{n},\sigma)$-equivalence coincide for skew $(\sigma,a)$-constacyclic codes via nonassociative Petit algebras over a commutative base ring $S$. It establishes a power-associativity criterion for monomial elements $\alpha t^k$, showing such elements satisfy $\alpha b = \sigma^{\mathbf{n}}(\alpha)\sigma^k(b)$, and then characterizes monomial homomorphisms $G_{\tau,\alpha,k}$ between Petit algebras, revealing severe restrictions for $k>1$ unless associativity arises. The results imply that in many nonassociative cases, all Hamming-weight-preserving isomorphisms are degree-one monomial maps, causing $(\mathbf{n},\sigma)$-isometry and $(\mathbf{n},\sigma)$-equivalence to coincide and enabling tighter code classifications; the work also corrects previous claims, provides counting formulas for isometry classes, and proposes refined notions of equivalence/isometry that capture all Hamming-preserving isomorphisms. The findings have implications for the classification of skew constacyclic codes and potential applications to quantum error-correcting code construction.
Abstract
We show that the notions of $(n,σ)$-isometry and $(n,σ)$-equivalence introduced by Ou-azzou et al coincide for most skew $(σ,a)$-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 $S[t;σ]/S[t;σ](t^n-a)$, which give rise to skew constacyclic codes.