Fractional Decoding of Algebraic Geometry Codes over Extension Fields

Core Concepts

Fractional decoding algorithms for algebraic geometry codes over extension fields using virtual projections and interleaved codes.

Abstract

The paper studies algebraic geometry codes from curves over the extension field Fqℓ through their virtual projections, which are algebraic geometric codes over the base field Fq. The virtual projections are used to provide fractional decoding algorithms for the codes over Fqℓ. Key highlights: Fractional decoding seeks to perform error correction using a smaller fraction of Fq-symbols than a typical decoding algorithm. The virtual projection of an algebraic geometry code can be seen as an algebraic geometry code itself, with the degree of a pole divisor of an annihilator function affecting the bound on the number of correctable errors. Viewing the virtual projections as interleaved codes allows, with high probability, correcting more errors than anticipated. The fractional decoding approaches can make use of advances in traditional decoding algorithms for algebraic geometry codes. The paper provides a general framework for fractional decoding of algebraic geometry codes over extension fields, with particular instances for important families of codes like those from Kummer extensions or Castle curves.

Stats

The number of correctable errors is bounded by: (n - (deg G + (l - m) max {deg(pt)∞: t ∈[m]})) / 2

Quotes

None.

Key Insights Distilled From

Fractional decoding of algebraic geometry codes over extension fields

by Eduardo Camp... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.07201.pdf

Deeper Inquiries

How can the virtual projection techniques be extended to other families of algebraic geometry codes beyond those considered in the paper

The virtual projection techniques discussed in the paper can be extended to other families of algebraic geometry codes by considering different partitions of evaluation points and associated annihilator functions. By carefully selecting the partition and the annihilator polynomials, it is possible to create virtual projections that can aid in error correction and decoding of a wide range of algebraic geometry codes. This approach can be applied to codes from various curves over different finite fields, allowing for the fractional decoding of codes beyond those specifically addressed in the paper. Additionally, by adapting the virtual projection method to suit the characteristics of different algebraic geometry codes, researchers can explore new avenues for error correction and decoding strategies in diverse code families.

What are potential limitations or drawbacks of the fractional decoding approach compared to traditional decoding of algebraic geometry codes over extension fields

While fractional decoding offers the advantage of performing error correction using a smaller fraction of symbols from the base field compared to traditional decoding algorithms, there are potential limitations and drawbacks to consider. One limitation is the dependency on the choice of partitions and annihilator functions, which may not always lead to successful error correction. The success of fractional decoding relies heavily on the accuracy of these choices, and if they are not optimal, the decoding performance may be suboptimal. Additionally, the complexity of determining the appropriate partitions and annihilator functions for different code families can be a challenging task, requiring significant computational resources and expertise. Moreover, the probabilistic nature of fractional decoding algorithms may result in occasional decoding failures, especially when the number of correctable errors is close to the decoding radius. This uncertainty can pose a limitation in practical applications where reliable error correction is crucial.

What are the implications of the fractional decoding framework for practical applications like distributed storage systems

The fractional decoding framework introduced in the paper has significant implications for practical applications like distributed storage systems. By enabling error correction using a reduced number of symbols from the base field, fractional decoding can help optimize storage efficiency and reduce network traffic in distributed storage environments. This approach allows for efficient data recovery and reconstruction with minimal resource utilization, making it well-suited for systems where bandwidth and storage capacity are limited. Additionally, the ability to correct errors using a smaller fraction of symbols can enhance the overall reliability and resilience of distributed storage systems, ensuring data integrity and availability even in the presence of errors or failures. Overall, the fractional decoding framework offers a promising solution for improving the performance and efficiency of distributed storage systems in real-world scenarios.