Locality Parameter Construction in Generalized AG Codes

Główne pojęcia

The author introduces a novel method to construct locally recoverable codes with a specific locality parameter in generalized AG codes, providing new insights into code construction and optimization.

Streszczenie

The content discusses the introduction of locality parameters in generalized AG codes for constructing locally recoverable codes. It explores constructions, comparisons with concatenated codes, and asymptotic studies using function fields.
In 1999, Xing, Niederreiter, and Lam introduced a generalization of AG codes using non-rational places. The paper presents a new approach to construct locally recoverable codes with specific locality parameters by utilizing non-rational places of degrees at most r. This method allows for the direct construction of codes whose dimension is not a multiple of the locality. An example reaching the Singleton-like bound for LRCs is provided along with an investigation into similarities with concatenated codes. The study includes explicit families of LRCs with locality 2 over finite fields of cardinality greater than 3 following this new approach.

Statystyki

d ≤ n − k − ⌈k/r⌉ + 2.
Several constructions of optimal LRCs exist.
Length of an optimal LRC is at most O(dq^3).
Rate of q-ary linear code C: k/n ≤ r/(r + 1).
Minimum distance d of C: d ≤ n - k - ⌈k/r⌉ + 2.
Gilbert-Varshamov-type bound for LRCs: Rq(r, δ) ≥ ...

Cytaty

"Our construction allows one to obtain directly codes whose dimension is not a multiple of the locality."
"An important difference is that our new approach from generalized AG-codes allows to construct directly codes whose dimension is not a multiple of the locality."

Kluczowe wnioski z

Introducing locality in some generalized AG codes

by Bastien Paci... o arxiv.org 03-04-2024

https://arxiv.org/pdf/2403.00430.pdf

Głębsze pytania

How does the proposed method compare to existing techniques in constructing locally recoverable codes

The proposed method of constructing locally recoverable codes using generalized AG codes introduces a new approach compared to existing techniques. By utilizing non-rational places of degrees at most r, the construction allows for the direct creation of codes with locality parameter r. This is a departure from traditional methods that often require additional steps to adjust the dimension to be a multiple of the locality. Additionally, this new method provides flexibility in choosing the degree of places used in encoding, allowing for more customization and optimization based on specific requirements.

What implications does this research have on distributed storage systems utilizing coding theory

This research has significant implications for distributed storage systems that utilize coding theory. Locally recoverable codes play a crucial role in such systems by enabling efficient data recovery processes when errors occur or data needs to be reconstructed. The ability to construct optimal LRCs with parameters reaching Singleton-like bounds opens up possibilities for improving fault tolerance and reliability in distributed storage environments. By achieving higher rates while maintaining low complexities, these optimized LRCs can enhance overall system performance and resilience against failures.

How can these findings be applied to other areas beyond coding theory

Beyond coding theory, the findings from this research can have applications in various other areas where error correction and fault tolerance are essential. For instance:

Communication Networks: The use of locally recoverable codes can improve network reliability by enabling efficient error detection and correction mechanisms.
Cloud Computing: Implementing optimized LRCs can enhance data integrity and availability in cloud storage services.
IoT Devices: Utilizing these advanced coding techniques can ensure robustness and resilience in IoT networks where connectivity issues may arise.
Data Security: Incorporating locally recoverable codes into encryption schemes can provide added protection against data loss or corruption.
By exploring these broader applications beyond coding theory, the research opens up avenues for enhancing system performance across various domains through improved error handling capabilities provided by locally recoverable codes.