Error Correction Decoding Algorithms of RS Codes Based on An Earlier Termination Algorithm to Find The Error Locator Polynomial (2407.19484v1)

Abstract: Reed-Solomon (RS) codes are widely used to correct errors in storage systems. Finding the error locator polynomial is one of the key steps in the error correction procedure of RS codes. Modular Approach (MA) is an effective algorithm for solving the Welch-Berlekamp (WB) key-equation problem to find the error locator polynomial that needs $2t$ steps, where $t$ is the error correction capability. In this paper, we first present a new MA algorithm that only requires $2e$ steps and then propose two fast decoding algorithms for RS codes based on our MA algorithm, where $e$ is the number of errors and $e\leq t$. We propose Improved-Frequency Domain Modular Approach (I-FDMA) algorithm that needs $2e$ steps to solve the error locator polynomial and present our first decoding algorithm based on the I-FDMA algorithm. We show that, compared with the existing methods based on MA algorithms, our I-FDMA algorithm can effectively reduce the decoding complexity of RS codes when $e<t$. Furthermore, we propose the $t_0$-Shortened I-FDMA ($t_0$-SI-FDMA) algorithm ($t_0$ is a predetermined even number less than $2t-1$) based on the new termination mechanism to solve the error number $e$ quickly. We propose our second decoding algorithm based on the SI-FDMA algorithm for RS codes and show that the multiplication complexity of our second decoding algorithm is lower than our first decoding algorithm (the I-FDMA decoding algorithm) when $2e<t_0+1$.

• The paper proposes modified algorithms for decoding Reed-Solomon (RS) codes that find the error locator polynomial using fewer steps, based on the actual number of errors.
• Algorithms like I-FDMA and t0-SI-FDMA significantly reduce multiplication complexity, showing gains up to 74.67% in specific test cases.
• These improved decoding efficiencies have practical implications for data storage systems and other applications relying on robust error correction.

An Insightful Overview of "Error Correction Decoding Algorithms of RS Codes Based on An Earlier Termination Algorithm to Find The Error Locator Polynomial"

The paper "Error Correction Decoding Algorithms of RS Codes Based on An Earlier Termination Algorithm to Find The Error Locator Polynomial" offers a meticulous examination of existing algorithms used for decoding Reed-Solomon (RS) codes, a valuable tool in error correction for storage systems. The authors propose a modification to the Modular Approach (MA) which traditionally requires $2t$ steps, where $t$ represents the error correction capability, to find the error locator polynomial. Instead, they introduce a modified MA that can find this polynomial in as few as $2e$ steps, where $e$ is the actual number of errors with $e \leq t$.

Overview of Contributions

1. Modified Modular Approach: The paper details a new Modular Approach that requires only $2e$ steps for identifying the error locator polynomial, enhancing computational efficiency when the number of errors $e$ is less than the correction capability $t$. This serves as a foundation for reducing decoding complexity significantly when $e < t$.
2. Improved Frequency Domain Modular Approach (I-FDMA): Building on the modified MA, the authors propose the I-FDMA algorithm, which efficiently computes the error locator polynomial through $2e$ iterations. This advancement claims a reduction in multiplication complexity ranging from 5.64% to 41.79% over prior methods when dealing with a moderate number of errors.
3. $t_0$-Shortened I-FDMA ($t_0$-SI-FDMA) Algorithm: This method introduces a predetermined even number $t_0$, less than $2t-1$, enabling quicker determination of $e$ within $2e$ iterations when $2e < t_0+1$. This approach further optimizes multiplication complexity by utilizing the early termination capability of the modified MA.
4. Two Decoding Algorithms: Using the I-FDMA algorithm as a foundation, the authors propose two decoding pathways for LCH-FFT-based RS codes. The first algorithm directly utilizes I-FDMA for computing polynomials, while the second combines $t_0$-SI-FDMA with an S-ESBM algorithm for improved performance when $e$ is significantly less than $t$.

Numerical Analysis and Results

The proposed methodologies were evaluated on RS codes with parameters $(n,k)=(256,224)$ and $(n,k)=(128,96)$. For an RS code with $t=16$, the I-FDMA algorithm achieved multiplication reduction of up to 74.67% in specific tests. The second algorithm, designed for small $e$, demonstrated a 22.49% to 59.22% reduction in multiplications, validating the effectiveness of these improvements.

Implications and Future Directions

This paper's implications are far-reaching in the field of data storage, particularly in systems where RS codes are utilized due to their robustness in error correction. By reducing computational steps and improving efficiency, these algorithms can facilitate faster data correction, which is critical in high-throughput environments like cloud storage and large data centers.

Theoretically, this work advances the understanding of error correction capability boundaries, challenging traditional assumptions about the complexity necessary for polynomial calculations in RS decoding.

For future research, exploring integration with other coding schemes or further optimization in conjunction with hardware advancements could provide additional efficiency enhancements. Moreover, generalization of these approaches to varied coding scenarios or more complex error patterns would deepen their applicability across different fields of industry and academia.

In conclusion, this research offers a robust contribution to the field of RS code decoding, providing both theoretical insights and practical tools for improved data error correction in diverse applications.