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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 correspondence, 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. In comparison to previous results based on Horner's rule, our hardware implementation for a (2720, 2550) shortened RS code over GF(2<sup>12</sup>) achieves much higher throughputs and better area-time complexity.

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