(σ, δ)-Polycyclic Codes in Ore Extensions Over Rings: Algebraic Structure Study

Keskeiset käsitteet

(σ, δ)-polycyclic codes in Ore extensions over rings have unique properties and relationships with other codes.

Tiivistelmä

この論文では、(σ, δ)-ポリサイクリックコードの代数構造に焦点を当てています。これらのコードは特定の環上で定義され、異なるアプローチから包括的な調査を提供します。BoucherとUlmerによって導入された概念を基に、これらのコードの性質や関係が探求されます。さらに、Euclidean dualsやHamming同型等価など、重要な概念が紹介されます。

Tilastot

R = 1
n = 1
f ∈ S
TC f = σ(v)C f + δ(v)
C f P = σ(P)AP−1 + δ(P)
Λ(g) := g(T) = Pn−1 i=0 giT i ∈ End(V, +)
TC(f)(v) := σ(v)Ct(f) - δ(v)

Lainaukset

"By studying (σ, δ)-polycyclic codes from several points of view, we achieve a unification of different approaches."
"(σ, δ)-polycyclic codes can provide improved distance bounds compared to skew cyclic codes."
"These terms represent one concept: the generalization of polycyclic codes into the non-commutative polynomial ring."
"(σ, δ)-polycyclic codes have unique properties and relationships with other codes."
"In this paper, we use the term “(σ, δ)-polycyclic codes” for this concept."

Tärkeimmät oivallukset

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

by Mary... klo arxiv.org 03-01-2024

https://arxiv.org/pdf/2312.07193.pdf

Syvällisempiä Kysymyksiä

How do (σ, δ)-polycyclic codes compare to traditional cyclic codes

(σ, δ)-polycyclic codes are a generalization of traditional cyclic codes. While cyclic codes are defined over commutative rings and generated by a single polynomial, (σ, δ)-polycyclic codes extend this concept to non-commutative rings using the Ore extension ring structure. This allows for more flexibility in encoding and decoding processes compared to traditional cyclic codes. Additionally, (σ, δ)-polycyclic codes encompass various generalizations of cyclic codes such as skew cyclic codes and polycyclic codes.

What are the practical applications of (σ, δ)-polycyclic codes in coding theory beyond theoretical studies

Beyond theoretical studies, (σ, δ)-polycyclic codes have practical applications in coding theory that can benefit various technological fields. These applications include error correction in communication systems like wireless networks and satellite communications where data integrity is crucial. They can also be utilized in cryptography for secure data transmission and storage. Furthermore, (σ, δ)-polycyclic codes play a role in quantum computing for creating efficient quantum error-correcting codes.

How can the findings on (σ, δ)-polycyclic codes be extended to other algebraic structures or mathematical domains

The findings on (σ, δ)-polycyclic codes can be extended to other algebraic structures or mathematical domains by exploring their connections with different types of rings and modules. For example:

Studying the properties of these codes over division rings or finite fields could provide insights into their behavior under different algebraic settings.
Investigating the relationship between (σ, δ)-polycyclic codes and other types of generalized linear block codes could lead to advancements in coding theory.
Exploring applications of these concepts in areas such as group theory or algebraic geometry may reveal new connections between coding theory and other branches of mathematics.