Using nonassociative algebras to classify skew polycyclic codes up to isometry and equivalence

Abstract: 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.