Reduced-Complexity Decoder of Long Reed-Solomon Codes Based on Composite Cyclotomic Fourier Transforms

TL;DR

This paper introduces a new partial composite cyclotomic Fourier transform-based decoder for long Reed-Solomon codes, significantly reducing computational complexity and enabling efficient hardware implementation with higher throughput.

Contribution

The paper proposes partial CCFTs for syndrome decoding of long RS codes, offering a modular, hardware-friendly approach that outperforms previous methods in complexity and throughput.

Findings

Significant reduction in computational complexity for long RS code decoding.

Hardware implementation achieves smaller gate count and higher throughput.

Demonstrated effectiveness on a (2720, 2550) RS code over GF(2^{12}).

Abstract

Long Reed-Solomon (RS) codes are desirable for digital communication and storage systems due to their improved error performance, but the high computational complexity of their decoders is a key obstacle to their adoption in practice. As discrete Fourier transforms (DFTs) can evaluate a polynomial at multiple points, efficient DFT algorithms are promising in reducing the computational complexities of syndrome based decoders for long RS codes. In this paper, we first propose partial composite cyclotomic Fourier transforms (CCFTs) and then devise syndrome based decoders for long RS codes over large finite fields based on partial CCFTs. The new decoders based on partial CCFTs achieve a significant saving of computational complexities for long RS codes. Since partial CCFTs have modular and regular structures, the new decoders are suitable for hardware implementations. To further verify and demonstrate the advantages of partial CCFTs, we implement in hardware the syndrome computation block for a $(2720, 2550)$ shortened RS code over GF$(2^{12})$. In comparison to previous results based on Horner's rule, our hardware implementation not only has a smaller gate count, but also achieves much higher throughputs.