Algebraic Soft Decoding Algorithms for Reed-Solomon Codes Using Module
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The interpolation based algebraic decoding for Reed-Solomon (RS) codes can correct errors beyond half of the code's minimum Hamming distance. Using soft information, algebraic soft decoding (ASD) can further enhance the performance. This paper introduces two ASD algorithms that utilize a new interpolation technique, the module minimization (MM). Achieving the desirable Gröbner basis, the MM interpolation is simpler than the existing Koetter's interpolation that is an iterative polynomial construction process. We will demonstrate how the MM technique can be utilized to solve the interpolation problem in the Koetter-Vardy decoding and the algebraic Chase decoding, respectively. Their re-encoding transformed variants will also be introduced. This transform reduces the size of module entries, leading to a reduced MM complexity. It is more effective for high rate codes. For an RS code of length n and dimension k, the MM complexity is O(n(n - k)l^5), where l is the y-degree of the interpolated polynomial. Re-encoding transform further reduces the MM complexity to O((n-k)^2l^5). The MM complexity characterizes complexity of the two ASD algorithms. Simulation results will be further provided to substantiate the proposals' decoding and complexity performances.
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