Constructing Minimal Linear Codes over F3 Using Characteristic and Ternary Functions

Alapfogalmak

This paper presents two new approaches to construct minimal linear codes of dimension n+1 over the finite field F3 using characteristic and ternary functions. The authors also obtain the weight distributions of these constructed minimal linear codes and show that a specific class of these codes violates the Ashikhmin-Barg condition.

Kivonat

The paper focuses on constructing minimal linear codes over the finite field F3.
Key highlights:

The authors define two types of functions - characteristic functions and ternary functions - using partial spreads in the vector space Fn3.
Using these functions, they construct two classes of minimal linear codes over F3 of dimension n+1.
The weight distributions of the constructed minimal linear codes are derived.
It is shown that a specific class of the constructed minimal linear codes violates the Ashikhmin-Barg condition, which provides a sufficient condition for a linear code to be minimal.
The authors prove that the constructed minimal linear codes satisfy the necessary and sufficient conditions for minimality.
For one class of the minimal linear codes, it is shown that the ratio of the minimum and maximum non-zero Hamming weights is less than 1/3, violating the Ashikhmin-Barg condition.

The paper provides a comprehensive analysis of the construction and properties of the minimal linear codes over F3, contributing to the understanding and development of this important class of error-correcting codes.

Statisztikák

The minimum and maximum non-zero Hamming weights of the constructed minimal linear codes are given by the following expressions:
wtmin = 2s(3^t - 1)
wtmax = 3^n - 3^(n-1) + 3^t - 2s

Idézetek

"In this paper, using the function defined in [9], we construct a class of minimal linear codes over F3, which satisfies wtmin/wtmax ≤ 1/3 < 2/3."
"Also we define new ternary function and construct minimal linear codes over F3, which violates Ashikhmin-Barg."

Főbb Kivonatok

A new approach to construct minimal linear codes over $\mathbb{F}_{3}$

by Wajid M. Sha... : arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.06768.pdf

Mélyebb kérdések

How can the construction of minimal linear codes over F3 be extended to higher dimensions beyond n+1

To extend the construction of minimal linear codes over F3 to higher dimensions beyond n+1, one approach could involve exploring more complex mathematical structures and functions. For instance, incorporating advanced algebraic techniques such as algebraic geometry or coding theory could provide insights into constructing minimal linear codes in higher dimensions. Additionally, leveraging properties of specific mathematical objects like algebraic curves or surfaces could offer new avenues for constructing minimal linear codes with increased dimensions. Furthermore, exploring the interplay between different mathematical structures and their applications in coding theory could lead to innovative methods for constructing minimal linear codes in higher dimensions over F3.

What are the potential applications of the minimal linear codes constructed in this paper that violate the Ashikhmin-Barg condition

The minimal linear codes constructed in this paper that violate the Ashikhmin-Barg condition have potential applications in cryptographic systems requiring specific error control capabilities. These codes could be utilized in scenarios where traditional error-correcting codes may not be suitable due to their violation of the Ashikhmin-Barg condition. One potential application could be in secure communication systems where the violation of the Ashikhmin-Barg condition could be leveraged to enhance the security and robustness of the communication protocol. Additionally, these codes could find applications in scenarios where non-standard error control mechanisms are required, such as in specialized data storage systems or secure data transmission protocols.

What insights from the analysis of these minimal linear codes over F3 can be applied to construct minimal linear codes over other finite fields

Insights from the analysis of minimal linear codes over F3, particularly those that violate the Ashikhmin-Barg condition, can be applied to construct minimal linear codes over other finite fields by adapting the construction techniques and mathematical principles used in the paper. By understanding the key characteristics and properties of these codes, researchers can explore how similar principles can be applied to construct minimal linear codes over different finite fields. Additionally, the analysis of weight distributions, necessary and sufficient conditions for minimality, and the impact of specific mathematical functions on code construction can serve as valuable guidelines for developing minimal linear codes over a variety of finite fields. This cross-disciplinary approach can lead to the discovery of novel methods for constructing minimal linear codes with diverse applications in various fields.