New Constructions of Galois Self-Orthogonal Algebraic Geometry Codes

Core Concepts

This paper presents a criterion for an algebraic geometry (AG) code to be Galois self-orthogonal (SO), and constructs new classes of (maximum distance separable) Galois SO AG codes from projective lines, elliptic curves, hyperelliptic curves, and Hermitian curves.

Abstract

The paper studies Galois self-orthogonal (SO) algebraic geometry (AG) codes, which are generalizations of Euclidean and Hermitian SO codes. The authors make the following key contributions:

They provide a criterion in Lemma 3.1 to determine if an AG code is Galois SO, answering the first part of the main problem.

Over projective lines, they propose an embedding method in Lemma 3.3 to construct more MDS Galois SO codes from known MDS Galois SO AG codes. They also present a new explicit construction of MDS Galois SO AG codes in Theorem 3.5.

Over projective elliptic curves, hyperelliptic curves, and Hermitian curves, the authors construct some new Galois SO AG codes with good parameters in Theorems 3.9, 3.10, and 3.11. These new (MDS) Galois SO AG codes are listed in Table 2.

The paper starts by reviewing basic concepts on algebraic function fields and AG codes. It then presents the main results on Galois SO AG codes, including the criterion, the embedding method, and the new constructions over different algebraic curves. The results provide a comprehensive solution to the main problem of constructing general Galois SO AG codes.

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On Galois self-orthogonal algebraic geometry codes

by Yun Ding,Shi... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2309.01051.pdf

Deeper Inquiries

What are some potential applications of the new Galois self-orthogonal algebraic geometry codes constructed in this paper

The new Galois self-orthogonal algebraic geometry codes constructed in this paper have several potential applications in various fields. One application is in cryptography, where error-correcting codes play a crucial role in securing data transmission and storage. These codes can be used to enhance the security and reliability of cryptographic systems by ensuring that data is transmitted and stored accurately. Additionally, these codes can be applied in network communications, satellite communications, and data storage systems to improve the efficiency and accuracy of data transmission. Furthermore, the construction of maximum distance separable (MDS) Galois self-orthogonal codes can be particularly beneficial in scenarios where maximum error-correcting capability is required, such as in critical communication systems and data centers.

How can the criterion and construction methods developed in this paper be extended to other types of algebraic curves beyond the ones considered

The criterion and construction methods developed in this paper can be extended to other types of algebraic curves beyond the ones considered by exploring different properties and characteristics of these curves. For instance, the criterion for an algebraic geometry (AG) code being Galois self-orthogonal can be adapted to apply to hyperelliptic curves, elliptic curves, and other types of algebraic curves. By analyzing the algebraic structures and properties of these curves, similar criteria can be established to identify Galois self-orthogonal codes. The construction methods can also be modified to accommodate the unique features of different algebraic curves, allowing for the creation of diverse classes of Galois self-orthogonal AG codes.

Are there any connections between the Galois self-orthogonal property of AG codes and their performance in practical coding and communication systems

The Galois self-orthogonal property of AG codes can have implications for their performance in practical coding and communication systems. The self-orthogonality property can enhance the error-correcting capabilities of AG codes, making them more resilient to noise and interference in communication channels. This property can lead to improved data transmission reliability and accuracy, especially in scenarios where data integrity is critical. Additionally, the self-orthogonal structure of these codes can simplify decoding algorithms and reduce computational complexity, resulting in more efficient communication systems. Overall, the Galois self-orthogonal property can contribute to the robustness and efficiency of AG codes in practical applications.