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For the majority of the applications of Reed-Solomon (RS) codes, harddecision decoding is based on syndromes. Recently, there has been renewedinterest in decoding RS codes without using syndromes. In this paper, weinvestigate the complexity of syndromeless decoding for RS codes, and compareit to that of syndrome-based decoding. Aiming to provide guidelines topractical applications, our complexity analysis differs in several aspects fromexisting asymptotic complexity analysis, which is typically based onmultiplicative fast Fourier transform (FFT) techniques and is usually in big Onotation. First, we focus on RS codes over characteristic-2 fields, over whichsome multiplicative FFT techniques are not applicable. Secondly, due tomoderate block lengths of RS codes in practice, our analysis is complete sinceall terms in the complexities are accounted for. Finally, in addition to fastimplementation using additive FFT techniques, we also consider directimplementation, which is still relevant for RS codes with moderate lengths.Comparing the complexities of both syndromeless and syndrome-based decodingalgorithms based on direct and fast implementations, we show that syndromelessdecoding algorithms have higher complexities than syndrome-based ones for highrate RS codes regardless of the implementation. Both errors-only anderrors-and-erasures decoding are considered in this paper. We also derivetighter bounds on the complexities of fast polynomial multiplications based onCantor's approach and the fast extended Euclidean algorithm.

Complexity Analysis of Reed-Solomon Decoding over GF(2^m) Without Using Syndromes