We propose new definitions of equivalence and isometry for skew polycyclic codes that will lead to tighter classifications than existing ones. This helps to reduce the number of previously known isometry and equivalence classes, and state precisely when these different notions coincide. In the process, we classify classes of skew $(f,\sigma,\delta)$-polycyclic codes with the same performance parameters, to avoid duplicating already existing codes. We exploit that the generator of a skew polycyclic code is in one-one correspondence with the generator of a principal left ideal in its ambient algebra. Algebra 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 codes. We allow the ambient algebras to be nonassociative, thus eliminating the need on restrictions on the length of the codes. The isometries between the ambient algebras can also be used to classify corresponding linear codes equipped with the rank metric.

翻译：我们针对斜多循环码提出了新的等价与等距定义，这些定义将产生比现有分类更精细的划分。这有助于减少先前已知的等距类与等价类的数量，并精确阐明这些不同概念在何种条件下重合。在此过程中，我们对具有相同性能参数的斜$(f,\sigma,\delta)$-多循环码类别进行分类，以避免重复已有的编码。我们利用斜多循环码的生成元与其环境代数中主左理想的生成元之间存在一一对应关系这一特性。保持汉明距离的代数同构（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，并保持编码的长度、维数与汉明距离。我们允许环境代数为非结合代数，从而消除了对编码长度的限制要求。环境代数间的等距映射亦可用于对配备秩度量的相应线性码进行分类。