Efficient Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes

Core Concepts

This paper constructs binary linear sum-rank-metric codes with matrix size 2×2 from quaternary BCH and Goppa codes, and presents fast decoding algorithms for these codes.

Abstract

The paper focuses on the construction and decoding of binary linear sum-rank-metric codes with matrix size 2×2.
Key highlights:

Constructs binary linear sum-rank-metric codes with matrix size 2×2 from quaternary BCH and Goppa codes. These codes are called BCH-type and Goppa-type sum-rank-metric codes.
Provides a reduction of the decoding in the binary sum-rank-metric space to the decoding in the Hamming-metric space. This reduction is applied to the BCH and Goppa-type codes.
Presents fast decoding algorithms for the BCH and Goppa-type binary linear sum-rank-metric codes, with complexity O(ℓ^2) operations in the field F4.
Constructs asymptotically good sequences of binary linear sum-rank-metric codes with matrix size 2×2 from Goppa codes, which can be decoded efficiently.
Compares the constructed codes with the existing sum-rank BCH codes, showing that the new codes have larger dimensions for the same minimum sum-rank distances.

Stats

wtsr(a2x + a1x^2) = 2wtH(a1) + 2wtH(a2) - 3|I|, where I = supp(a1) ∩ supp(a2).
The minimum sum-rank distance of SR(C1, C2) is at least max{min{d1, 2d2}, min{d2, 2d1}}.

Quotes

"Sum-rank-metric codes have wide applications in multishot network coding, see [35, 43, 47], space-time coding, see [50], and coding for distributed storage, see [14, 33, 36]."
"Fast algebraic decoding and list-decoding algorithms for BCH codes, Goppa codes and algebraic geometry codes in the Hamming-metric are well-developed, see [27, Chapter 5], [16,19] and [24]."

Key Insights Distilled From

Construction and Fast Decoding of Binary Linear Sum-Rank-Metric Codes

by Hao Chen,Yan... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2311.03619.pdf

Deeper Inquiries

What other classes of Hamming-metric codes can be used to construct efficient binary linear sum-rank-metric codes with matrix size 2×2

In addition to BCH and Goppa codes, other classes of Hamming-metric codes that can be used to construct efficient binary linear sum-rank-metric codes with matrix size 2×2 include Reed-Solomon codes and algebraic geometry codes. These codes have well-developed decoding algorithms and can be adapted to the sum-rank metric space. By leveraging the properties and structures of these codes, it is possible to construct larger and more efficient binary linear sum-rank-metric codes with matrix size 2×2.

How can the decoding algorithms for the constructed sum-rank-metric codes be further improved in terms of complexity

To further improve the decoding algorithms for the constructed sum-rank-metric codes in terms of complexity, advanced techniques such as parallel processing, optimized data structures, and algorithmic optimizations can be implemented. By utilizing parallel computing capabilities, the decoding process can be distributed across multiple processors or cores, reducing the overall decoding time. Additionally, implementing efficient data structures and algorithms tailored to the specific characteristics of sum-rank-metric codes can help streamline the decoding process and reduce computational complexity. By continuously refining and optimizing the decoding algorithms, it is possible to achieve faster and more efficient decoding of binary linear sum-rank-metric codes with matrix size 2×2.

What are the potential applications of the efficient binary linear sum-rank-metric codes with matrix size 2×2 in areas such as network coding and distributed storage

The efficient binary linear sum-rank-metric codes with matrix size 2×2 have various potential applications in areas such as network coding and distributed storage. In network coding, these codes can be used for error correction and data transmission in multi-shot network scenarios. By efficiently encoding and decoding data using these codes, network performance can be improved in terms of reliability and throughput. In distributed storage systems, the codes can be utilized for data storage and retrieval, ensuring data integrity and availability across distributed storage nodes. The efficient decoding algorithms for these codes enable quick and accurate data recovery, enhancing the overall reliability and efficiency of distributed storage systems.