Symbol-Based Belief Propagation Decoding of Reed-Solomon Codes

We propose a symbol-based belief propagation (BP) algorithm for iterative soft-decision decoding of Reed-Solomon (RS) codes. Complexity reduction is achieved by using a fast Fourier transform (FFT)-based BP algorithm. Parity-check matrix adaptation based on the reliability of the codeword symbols is an essential step to make the BP algorithm effective on high-density parity-check matrices characteristic of RS codes. The matrix adaptation, as well as all other operations, is performed at symbol level such that bit-to-symbol and symbol-to- bit conversions are avoided. A moderate coding gain over algebraic hard-decision decoding is achieved on additive white Gaussian noise channels. The symbol-based algorithm presented in this paper addresses the main weakness of its bit-based counterpart, namely the prohibitively high complexity for practical applications that use long codes and large finite fields, albeit with less coding gain.