Employing isomorphisms between their ambient rings, we propose new definitions of equivalence and isometry for skew polycyclic codes that will lead to tighter classifications than existing ones. This reduces the number of previously known isometry and equivalence classes. In the process, we classify classes of skew $(f,σ,δ)$-polycyclic codes with the same performance parameters, to avoid duplicating already existing codes, and state precisely when different notions of equivalence coincide. The generator of a skew polycyclic code is in one-one correspondence with the generator of a principal left ideal in its nonassociative unital ambient ring. By allowing the ambient rings to be nonassociative, we eliminate the need on restrictions on the length of the codes. Ring isomorphisms that preserve the Hamming distance (called isometries) map generators of principal left ideals to generators of principal left ideals and preserve length, dimension, and Hamming distance of the corresponding isometric skew polycyclic codes.

翻译：通过利用其环境环之间的同构，我们提出了斜多环码的等价与等距新定义，这将产生比现有分类更精细的划分。这减少了先前已知的等距与等价类数量。在此过程中，我们分类了具有相同性能参数的斜$(f,σ,δ)$-多环码类别，以避免重复已有的编码，并精确阐述了不同等价概念何时重合。斜多环码的生成元与其非结合幺元环境环中的主左理想生成元一一对应。通过允许环境环为非结合结构，我们消除了对码长的限制。保持汉明距离的环境环同构（称为等距映射）将主左理想的生成元映射为主左理想的生成元，并保持对应等距斜多环码的码长、维数与汉明距离。