Characterization and Asymptotic Properties of Galois Self-Dual 2-Quasi Constacyclic Codes over Finite Fields

Core Concepts

The paper characterizes the algebraic structure of 2-quasi λ-constacyclic codes over finite fields and their Galois self-dual properties. It also investigates the asymptotic goodness of Hermitian and Euclidean self-dual 2-quasi λ-constacyclic codes.

Abstract

The paper studies 2-quasi λ-constacyclic codes over finite fields F with cardinality q = pℓ, where p is a prime and ℓ is a positive integer. It characterizes the algebraic structure of these codes and their Galois self-dual properties:

It shows that any 2-quasi λ-constacyclic code C over F can be written as (C1 × C2) ⊕ Cb,bg, where C1, C2, Cb are ideals of the quotient algebra Rλ = F[X]/(Xn - λ) and g is a unit in Cb.

It provides necessary and sufficient conditions for a 2-quasi λ-constacyclic code to be Galois self-dual, depending on whether λ1+ph = 1 or not.

It proves that if λ1+ph ≠ 1, then the Galois self-dual 2-quasi λ-constacyclic codes are asymptotically bad.

When ℓ is even and λ1+pℓ/2 = 1, it shows that the Hermitian self-dual 2-quasi λ-constacyclic codes are asymptotically good.

When pℓ ≠ 3 (mod 4) and λ2 = 1, it proves that the Euclidean self-dual 2-quasi λ-constacyclic codes are asymptotically good.

The paper introduces a new operator "∗" on the quotient ring Rλ, which becomes a useful technique for studying the Galois duality property of λ-constacyclic codes.

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Key Insights Distilled From

Galois Self-dual 2-quasi Constacyclic Codes over Finite Fields

by Yun Fan,Yue ... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08402.pdf

Deeper Inquiries

What are some potential applications of Galois self-dual 2-quasi constacyclic codes in practice

Galois self-dual 2-quasi constacyclic codes have various potential applications in practice. One application is in the field of cryptography, where error-correcting codes play a crucial role in ensuring secure communication. These codes can be used to protect data transmission against noise and interference, making them essential in secure communication protocols. Additionally, these codes can also be applied in data storage systems to enhance data reliability and integrity. By utilizing the properties of Galois self-dual codes, data can be stored and retrieved efficiently with minimal errors.

How can the techniques developed in this paper be extended to study the asymptotic properties of other families of constacyclic codes

The techniques developed in the paper can be extended to study the asymptotic properties of other families of constacyclic codes by adapting the methodology to different code structures. By applying similar algebraic characterizations and duality properties to other families of constacyclic codes, researchers can analyze their asymptotic behavior and determine their performance in various applications. This extension can provide insights into the efficiency and reliability of different constacyclic code constructions, leading to advancements in coding theory and applications.

Are there any connections between the asymptotic behavior of Galois self-dual 2-quasi constacyclic codes and the distribution of primitive roots modulo the field size

There is a potential connection between the asymptotic behavior of Galois self-dual 2-quasi constacyclic codes and the distribution of primitive roots modulo the field size. The properties of primitive roots in finite fields can impact the algebraic structure and duality properties of constacyclic codes. By studying the distribution of primitive roots and their relationship with code constructions, researchers can gain a deeper understanding of the performance and characteristics of constacyclic codes. This connection can lead to further insights into the asymptotic properties of codes and their applications in various fields.