A Family of Self-Orthogonal Divisible Linear Codes with Locality 2

Core Concepts

This paper constructs a family of self-orthogonal divisible linear codes with locality 2 using the trace and norm functions over finite fields. The codes are shown to have only three, four or five nonzero weights, and several optimal or almost optimal linear codes and locally recoverable codes are derived.

Abstract

The paper presents a construction of self-orthogonal divisible linear codes with locality 2 using the trace and norm functions over finite fields. The key highlights and insights are: The authors use the trace and norm functions over finite fields to construct a family of linear codes. The weight distributions of the codes are determined in three cases via Gaussian sums. The codes are shown to be self-orthogonal divisible codes with only three, four or five nonzero weights in these cases. The authors prove that this family of linear codes has locality 2. Several optimal or almost optimal linear codes and locally recoverable codes are derived from this construction. In particular, an infinite family of distance-optimal binary linear codes with respect to the sphere-packing bound is obtained. The self-orthogonal codes derived in this paper can be used to construct lattices and have nice applications in distributed storage.

Stats

The paper provides the following key metrics and figures: The length of the code CD is given as (qm - 1)(qm2 - q) / q(qm2 - 1) + 1. The weight distribution of CD is provided in three cases, with the frequencies of the nonzero weights given.

Quotes

"The codes are shown to be self-orthogonal divisible codes with only three, four or five nonzero weights in these cases." "Several optimal or almost optimal linear codes and locally recoverable codes are derived." "In particular, an infinite family of distance-optimal binary linear codes with respect to the sphere-packing bound is obtained."

Key Insights Distilled From

A family of self-orthogonal divisible codes with locality 2

by Ziling Heng,... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18437.pdf

Deeper Inquiries

How can the construction of self-orthogonal divisible linear codes with locality 2 be extended to other finite field parameters or code properties

The construction of self-orthogonal divisible linear codes with locality 2 can be extended to other finite field parameters by considering different values for the integers m, m1, and m2. By varying these parameters, we can explore the impact on the weight distribution, code properties, and the overall performance of the codes. Additionally, the construction can be adapted to different field sizes and characteristics, allowing for a broader range of applications and theoretical investigations. To extend the construction to other code properties, one could explore variations in the weight distribution, investigate different weight enumerators, or consider alternative methods for determining the locality of the codes. By tweaking the construction method and parameters, it is possible to create a diverse set of self-orthogonal divisible codes with locality 2 that exhibit unique properties and behaviors.

What are the potential applications of these self-orthogonal divisible codes with locality 2 beyond distributed storage, such as in cryptography or quantum computing

The self-orthogonal divisible codes with locality 2 derived in this study have various potential applications beyond distributed storage. In the field of cryptography, these codes can be utilized for secure communication, error detection, and correction in cryptographic protocols. The self-orthogonality property enhances the security and reliability of the codes, making them suitable for cryptographic applications where data integrity is crucial. In quantum computing, these codes can play a significant role in error correction and fault tolerance. Quantum codes based on self-orthogonal divisible codes with locality 2 can help mitigate errors that arise due to noise and decoherence in quantum systems. By leveraging the unique properties of these codes, quantum computations can be made more robust and error-resistant, paving the way for advancements in quantum information processing. Furthermore, these codes may find applications in other areas such as network coding, distributed computing, and error-resilient communication systems. Their ability to efficiently detect and correct errors while maintaining a low decoding complexity makes them valuable in scenarios where data reliability and efficiency are paramount.

Can the techniques used in this paper be applied to construct other families of linear codes with desirable properties, such as high rate or low complexity decoding

The techniques employed in constructing self-orthogonal divisible codes with locality 2 can be adapted to create other families of linear codes with desirable properties. By exploring different functions over finite fields, utilizing trace and norm operations, and investigating weight distributions through Gaussian sums, researchers can develop a wide range of linear codes with specific characteristics. For instance, the approach of using trace and norm functions can be extended to construct codes with different locality values, diverse weight distributions, or specific error-correcting capabilities. By modifying the defining sets, adjusting the parameters, or exploring alternative functions, new families of linear codes can be designed to meet specific requirements in various applications. Additionally, the insights gained from studying self-orthogonal divisible codes with locality 2 can inspire the development of novel coding schemes, optimal codes, or codes with enhanced performance metrics. By building upon the foundations laid out in this research, researchers can continue to innovate in the field of coding theory and contribute to advancements in communication, storage, and computing technologies.