(σ, δ)-Polycyclic Codes in Ore Extensions Over Rings

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

(σ, δ)-polycyclic codes differ from traditional cyclic codes in that they are defined as submodules in the quotient module of an Ore extension ring, where the ring is finite but not necessarily commutative. Traditional cyclic codes, on the other hand, are defined over commutative rings and are generated by a single polynomial. In contrast, (σ, δ)-polycyclic codes are a generalization that includes various types of cyclic codes, such as skew cyclic codes, skew negacyclic codes, and polycyclic codes. The algebraic structure of (σ, δ)-polycyclic codes allows for a more flexible and powerful encoding and decoding process compared to traditional cyclic codes.

Q: What implications do the findings have for practical applications in coding theory

The findings regarding (σ, δ)-polycyclic codes have significant implications for practical applications in coding theory. By studying these codes from various perspectives, researchers can unify different approaches and provide a comprehensive survey of various generalizations of cyclic codes. This unification can lead to the development of more efficient and robust coding schemes for data transmission and storage. Additionally, the classification of isometrically equivalent codes, such as Hamming isometry equivalence, can help in optimizing error correction capabilities and improving the overall performance of coding systems in practical applications.

Q: How can the concept of Hamming isometry equivalence be extended to other types of codes

The concept of Hamming isometry equivalence can be extended to other types of codes beyond (σ, δ)-polycyclic codes. By defining appropriate isomorphisms that preserve the Hamming weight of codewords, researchers can explore the equivalence between different types of codes, such as linear block codes, convolutional codes, and algebraic geometric codes. Extending the concept of Hamming isometry equivalence to a broader range of codes can facilitate the comparison and analysis of coding schemes in various communication systems and applications. This extension can lead to the development of more versatile and interoperable coding techniques in the field of coding theory.

Core Concepts

(σ, δ)-Polycyclic codes in Ore extensions over rings are studied, revealing their properties and relationships.

Abstract

The paper explores (σ, δ)-polycyclic codes in Ore extension rings.
It establishes relationships between Euclidean duals and (σ, δ)-sequential codes.
The study delves into the concept of Hamming isometry equivalence.
The (σ, δ)-Mattson-Solomon transform and decomposition of codes are discussed.
The structure of the paper is outlined in sections focusing on different aspects of (σ, δ)-polycyclic codes.

Stats

An Ore extension is a non-commutative polynomial ring.
(σ, δ)-polycyclic codes are defined as ideals in the quotient ring.
The Euclidean duals of (σ, δ)-polycyclic codes are (σ, δ)-sequential codes.

Quotes

"The study establishes relationships between Euclidean duals and (σ, δ)-sequential codes."
"The paper delves into the concept of Hamming isometry equivalence."

Key Insights Distilled From

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

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

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

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

Deeper Inquiries

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

(σ, δ)-polycyclic codes differ from traditional cyclic codes in that they are defined as submodules in the quotient module of an Ore extension ring, where the ring is finite but not necessarily commutative. Traditional cyclic codes, on the other hand, are defined over commutative rings and are generated by a single polynomial. In contrast, (σ, δ)-polycyclic codes are a generalization that includes various types of cyclic codes, such as skew cyclic codes, skew negacyclic codes, and polycyclic codes. The algebraic structure of (σ, δ)-polycyclic codes allows for a more flexible and powerful encoding and decoding process compared to traditional cyclic codes.

What implications do the findings have for practical applications in coding theory

The findings regarding (σ, δ)-polycyclic codes have significant implications for practical applications in coding theory. By studying these codes from various perspectives, researchers can unify different approaches and provide a comprehensive survey of various generalizations of cyclic codes. This unification can lead to the development of more efficient and robust coding schemes for data transmission and storage. Additionally, the classification of isometrically equivalent codes, such as Hamming isometry equivalence, can help in optimizing error correction capabilities and improving the overall performance of coding systems in practical applications.

How can the concept of Hamming isometry equivalence be extended to other types of codes

The concept of Hamming isometry equivalence can be extended to other types of codes beyond (σ, δ)-polycyclic codes. By defining appropriate isomorphisms that preserve the Hamming weight of codewords, researchers can explore the equivalence between different types of codes, such as linear block codes, convolutional codes, and algebraic geometric codes. Extending the concept of Hamming isometry equivalence to a broader range of codes can facilitate the comparison and analysis of coding schemes in various communication systems and applications. This extension can lead to the development of more versatile and interoperable coding techniques in the field of coding theory.

(σ, δ)-Polycyclic Codes in Ore Extensions Over Rings