Infinite Classes of Constacyclic Codes with Excellent Minimum Distance Properties

Core Concepts

Two infinite classes of constacyclic codes are constructed and analyzed, one of which contains ternary negacyclic self-dual codes with very good minimum distance properties.

Abstract

The paper presents two main contributions:

Two infinite classes of constacyclic codes C are constructed such that the minimum distances d(C) and d(C⊥) both have very good lower bounds.

Two infinite classes of ternary negacyclic self-dual codes with square-root-like lower bounds on their minimum distances are obtained.

The first class of constacyclic codes C(1,n) are ternary negacyclic self-dual codes with excellent parameters. For example, when m=4, C(1,40) is a ternary [40,20,9] self-dual code, which has better minimum distance than the best known ternary self-dual [40,20,12] code.
The second class of constacyclic codes C(q,m,ℓ) are analyzed in detail. Their dimensions and minimum distances are lower bounded. It is shown that for certain parameters, these codes have optimal or best known parameters.
The key techniques used include:

Splitting the defining set Ω(1)(r,n) based on Hamming weight to construct the first class of negacyclic codes.
Leveraging the structure of projective Reed-Muller codes to construct and analyze the second class of constacyclic codes.
Applying the constacyclic BCH bound to lower bound the minimum distances.

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Key Insights Distilled From

Two classes of constacyclic codes with a square-root-like lower bound

by Tingfang Che... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18438.pdf

Deeper Inquiries

What other classes of constacyclic codes with good parameters can be constructed using similar techniques

In addition to the classes of constacyclic codes constructed in the provided context, similar techniques can be used to construct other classes of constacyclic codes with good parameters. For example, by varying the choice of primitive elements and the properties of the finite fields, new classes of constacyclic codes can be generated. Additionally, exploring different factorizations of the polynomial equations and applying specific algebraic structures can lead to the creation of constacyclic codes with desirable properties. By leveraging the principles of constacyclicity and the underlying algebraic structures, researchers can continue to discover new classes of constacyclic codes with optimal parameters.

Can the techniques developed in this work be extended to construct constacyclic codes over other finite fields or with different lengths

The techniques developed in this work can be extended to construct constacyclic codes over other finite fields or with different lengths. By adapting the construction methods and considering the properties of different finite fields, researchers can explore the generation of constacyclic codes in various settings. For instance, investigating constacyclic codes over extension fields or prime fields can lead to the discovery of new classes of codes with unique characteristics. Moreover, varying the lengths of the codes and exploring the impact on the parameters can provide insights into the behavior of constacyclic codes in different contexts.

Are there applications of these constacyclic codes in areas like coding theory, cryptography, or communications that are worth exploring

Constacyclic codes have applications in various fields such as coding theory, cryptography, and communications. In coding theory, constacyclic codes are utilized for error detection and correction in data transmission systems. Their algebraic properties make them suitable for efficient encoding and decoding processes, leading to reliable communication systems. In cryptography, constacyclic codes play a role in secure data transmission and encryption schemes. By leveraging the properties of constacyclic codes, cryptographic protocols can be designed to ensure data confidentiality and integrity. In communications, constacyclic codes are used in wireless communication systems, satellite communication, and network protocols to enhance data reliability and transmission efficiency. Exploring the applications of constacyclic codes in these areas can lead to advancements in technology and improved system performance.