$(σ,δ)$-polycyclic codes in Ore extensions over rings

TL;DR

This work develops a comprehensive framework for $(\sigma,\delta)$-polycyclic codes over finite, potentially noncommutative rings by framing them as $S$-submodules of $S/Sf$ with the Ore extension $S=R[x,\sigma,\delta]$. It establishes duality results via a non-degenerate sesquilinear form and an annihilator construction, ensuring duals remain $(\sigma,\delta)$-polycyclic, and it classifies Hamming isometric equivalence through monomial transformations. The paper then leverages Wedderburn polynomials to define simple-root codes and builds a $(\sigma,\delta)$-Mattson-Solomon transform, enabling a Vandermonde-based decomposition of these codes according to eigenvalues of a $(\sigma,\delta)$-PLT. Finally, it presents a direct-sum decomposition of simple-root codes and discusses both theoretical and potential practical implications for code construction, with plans to extend to multivariable settings and optimal-code design.

Abstract

In this paper, we study the algebraic structure of $(σ,δ)$-polycyclic codes, defined as submodules in the quotient module $S/Sf$, where $S=R[x,σ,δ]$ is the Ore extension ring, $f\in S$, and $R$ is a finite but not necessarily commutative ring. We establish that the Euclidean duals of $(σ,δ)$-polycyclic codes are $(σ,δ)$-sequential codes. By using $(σ,δ)$-Pseudo Linear Transformation, we define the annihilator dual of $(σ,δ)$-polycyclic codes. Then, we demonstrate that the annihilator duals of $(σ,δ)$-polycyclic codes maintain their $(σ,δ)$-polycyclic nature. Furthermore, we classify when two $(σ,δ)$-polycyclic codes are Hamming isometrical equivalent. By employing Wedderburn polynomials, we introduce simple-root $(σ,δ)$-polycyclic codes. Subsequently, we define the $(σ, δ)$-Mattson-Solomon transform for this class of codes and we address the problem of decomposing these codes by using the properties of Wedderburn polynomials.

Figures (2)

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Theorems & Definitions (64)

• Proposition 1: evaluation Lemma 2 and LEROY Proposition 1.2
• Corollary 1: LEROY Corollary 1.3
• Lemma 1: evaluation Corollary 2.10
• Definition 1
• Remark 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4