Progressive Module Minimization for Re-encoding Transformed Soft Decoding of RS Codes

Abstract

The interpolation based algebraic decoding for Reed-Solomon (RS) codes can correct errors beyond half of the code’s minimum Hamming distance through constructing a minimum polynomial Q(x,y) and finding its y-roots. The progressive algebraic soft decoding (PASD) constructs Q(x, y) with a progressively enlarged y-degree and terminates once the message is decoded, adapting the decoding capability and computation to the channel. This paper proposes the re-encoding transformed PASD algorithm, in which Q(x, y) is progressively constructed by the low-complexity module minimization (MM) technique. Re-encoding transform (ReT) results in a common divisor for polynomials of the image of the submodule basis. It can be removed, leading to a simpler image expansion and reduction. Consequently, Q(x,y) is constructed through the isomorphic image of the progressively enlarged submodule basis. Our complexity analysis characterizes the complexity reduction brought by the transform and shows high rate codes benefit a greater complexity reduction.