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

TL;DR

This paper introduces new classifications of skew polycyclic codes using nonassociative algebraic structures, leading to more precise equivalence and isometry distinctions and reducing redundant code classifications.

Contribution

It proposes novel definitions of code equivalence and isometry based on ambient ring isomorphisms, utilizing nonassociative rings to improve classification accuracy.

Findings

Reduced the number of known isometry and equivalence classes.

Classified skew polycyclic codes with identical performance parameters.

Established conditions when different notions of equivalence coincide.

Abstract

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 $(f,\sigma,\delta)$-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.