A low complexity iterative technique for soft decision decoding of reed-solomon codes
ICC'09 Proceedings of the 2009 IEEE international conference on Communications
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
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Two approaches to soft decoding of Reed-Solomon Codes (of block length n) are presented. The first procedure is an algebraic, Chase-type, technique which first produces a set of test-vectors that are equivalent on all except h≪n coordinate positions. The similarity of the test-vectors is utilized to reduce the complexity of interpolation, the process of constructing a set of polynomials that obey constraints imposed by each test-vector. By first considering the equivalent indices, a polynomial common to all test-vectors is constructed. The required set of polynomials is then produced by interpolating the final η dissimilar indices utilizing a binary-tree structure. In the factorization step, a candidate message is extracted from each interpolation polynomial. Although an expression for the direct evaluation of each candidate message is provided, carrying-out this calculation for each polynomial is extremely complex. Thus, a novel, reduced-complexity, methodology is also given. Although suboptimal, simulation results affirm the loss in performance incurred by this procedure is decreasing with code length, and negligible for long (n 100) codes. Significant coding gains are shown to be achievable over traditional hard-decision decoding procedures at a comparable computational complexity. These gains are, furthermore, shown to be similar to recently proposed algebraic techniques that bear a significantly more complex implementation than the proposed algorithm. The second proposed technique is an iterative methodology that applies the output of subsequent iterations of Belief-Propagation (BP) as input to a legacy decoding algorithm. However, due to the suboptimal performance of BP conducted on the inherently dense Reed-Solomon parity-check matrix, a method is provided to construct reduced-density, binary, parity-check equations. Iterative decoding is then implemented by (1) utilizing a subset of a redundant set of parity-check equations to minimize the number of connections into the least-reliable bits or (2) augmenting the parity-check matrix with phantom-checks that, when added to existing rows, maximally improve the structure of the resulting graph. Simulation results show that performance comparable to the best known Reed-Solomon decoding techniques is achievable with these methodologies. However, unlike these existing procedures, the architecture of the proposed algorithm allows for a practical implementation in hardware.