Progressive Algebraic Soft-Decision Decoding of Reed–Solomon Codes Using Module Minimization

The algebraic soft-decision decoding (ASD) of Reed–Solomon (RS) codes yields a competent decoding performance with a polynomial-time complexity. But its complexity remains high due to the interpolation that generates the interpolation polynomial
$Q(x,y)$
. The progressive ASD (PASD) algorithm has been introduced to construct
$Q(x,y)$
with a progressively enlarged
$y$
-degree, adjusting its error-correction capability and computation to the received information. However, this progressive decoding is realized at the cost of memorizing the intermediate decoding information. To overcome this challenge, this paper proposes a new PASD algorithm which is evolved from the ASD using module minimization (MM) interpolation. Polynomial
$Q(x,y)$
can be constructed through the image of the progressively enlarged submodule basis without the need of memorizing the intermediate decoding information, eliminating the memory cost of progressive decoding. The MM interpolation also attributes to a remarkably lower complexity than the original PASD algorithm. Furthermore, a complexity reducing variant is proposed based on assessing the degree of Lagrange interpolation polynomials. We also analyze the complexity of the proposed decoding methods and reveal their channel dependent feature. Our simulation results show their low-complexity and advanced decoding performances.