Improved Upper Bounds on the Number of Non-Zero Weights of Constacyclic Codes

Core Concepts

The paper establishes improved upper bounds on the number of non-zero weights of simple-root constacyclic codes by calculating the number of orbits of a larger subgroup of the automorphism group of the code.

Abstract

The paper focuses on establishing improved upper bounds on the number of non-zero weights of simple-root constacyclic codes over finite fields.
Key highlights:

The authors choose a larger subgroup G'' of the automorphism group Aut(C) of the constacyclic code C, which contains the subgroup G' used in previous work.
By calculating the number of G''-orbits of C{0}, the authors derive an explicit upper bound on the number of non-zero weights of C that is strictly smaller than the previous upper bounds.
The results generalize and improve upon the main results in prior work, removing certain constraints and providing a new method to construct few-weight constacyclic codes.
For two special classes of constacyclic codes, the authors obtain even smaller upper bounds by replacing G'' with larger subgroups of the automorphism groups.
Several examples are presented to demonstrate that the new upper bounds are tight and better than the previous ones.

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New upper bounds on the number of non-zero weights of constacyclic codes

by Li Chen,Yuqi... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00309.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to obtain improved upper bounds for other families of linear codes beyond constacyclic codes

The techniques developed in the paper for obtaining improved upper bounds on the number of non-zero weights of constacyclic codes can be extended to other families of linear codes by considering different automorphism groups and their actions on the codes. For example, one could explore the use of larger subgroups of the automorphism group to derive tighter upper bounds for cyclic codes, negacyclic codes, and other families of linear codes. By carefully analyzing the group actions and orbits of non-zero elements, similar results can be achieved for different types of linear codes. Additionally, the concept of group actions and orbits can be applied to non-constacyclic codes to establish upper bounds on the number of non-zero weights in those codes as well.

Can the authors' approach be adapted to derive lower bounds on the number of non-zero weights of constacyclic codes

The authors' approach can be adapted to derive lower bounds on the number of non-zero weights of constacyclic codes by considering the structure of the automorphism group and its subgroups. By analyzing the group actions and orbits of non-zero elements in the code, one can determine the minimum number of distinct weights that must exist in the code. This can be achieved by looking at the properties of the automorphisms and their interactions with the code elements. By carefully examining the group orbits and the conditions for equality in the upper bound results, one can infer constraints that lead to lower bounds on the number of non-zero weights in constacyclic codes.

What are some potential applications of few-weight constacyclic codes constructed using the authors' main results

Few-weight constacyclic codes constructed using the authors' main results have several potential applications in coding theory and practical communication systems. One application is in the design of efficient error-correcting codes for data transmission over noisy channels. Few-weight codes with a small number of non-zero weights are desirable for reducing decoding complexity and improving error correction capabilities. These codes can be used in applications where computational resources are limited or where fast decoding is essential.
Another application is in cryptography, where few-weight codes can be utilized in secret sharing schemes and cryptographic protocols. By constructing constacyclic codes with a small number of non-zero weights, it is possible to enhance the security and efficiency of cryptographic systems. Few-weight codes can also be employed in authentication codes and secure communication protocols to ensure data integrity and confidentiality. Overall, the construction and analysis of few-weight constacyclic codes have significant implications for various areas of information theory and coding applications.