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

TL;DR

This work addresses classifying skew polycyclic codes up to isometry and equivalence by embedding them in nonassociative Petit algebras and using isomorphisms that preserve Hamming weight, thereby refining prior classifications. The authors define monomial (degree $k$) isomorphisms $G_{\tau,\alpha,k}$ and Chen isometries $G_{id,\alpha,k}$ to transfer generators between ambient algebras $R/Rf$ and $R/Rh$, and establish necessary and sufficient conditions (notably $\tau(a_i)=N_{m-i}^{\sigma}(\sigma^i(\alpha))b_i$) for equivalence or isometry. They derive explicit equivalence classes, provide criteria to distinguish equivalent/non-equivalent codes, and specialize results to finite fields to count Chen isometry/equivalence classes of skew constacyclic codes via the norm maps $N_m^{\sigma}$. The framework yields tighter partitions of code classes, facilitates deduplication, and offers practical tools for searching good codes and potential quantum-code constructions, with future work on duality and nonzero-$\delta$ cases. Throughout, the approach relies on the $S_f$ Petit algebra perspective and the interplay of automorphisms, norms, and monomial isomorphisms.

Abstract

Employing isomorphisms between their ambient algebras, 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 ambient algebra. We allow the ambient algebras to be nonassociative, thus eliminating the need on restrictions on the length of the codes. 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. The isometries between the ambient algebras can also be used to classify corresponding linear codes equipped with the rank metric.

Theorems & Definitions (72)

• Theorem
• Theorem
• Theorem 2.1
• Proposition 2.2
• Proposition 2.3
• Example 2.4
• Lemma 2.5
• Definition 2.6
• Theorem 2.7
• Theorem 2.8