Multiplicity assignments for algebraic soft-decoding of reed-solomon codes

Abstract

A soft-decision decoding algorithm for ReedSolomon codes was recently proposed in [2]. This algorithm converts probabilities observed at the channel output into algebraic interpolation conditions, specified in terms of a multiplicity matrix M. Koetter-Vardy [2] show that the probability of decoding failure is given by h { S M 5 A ( M ) } , where SM is a random variable and A ( M ) is a known function of M . They then compute the multiplicity matrix M that maximizes the e q e c t e d value of SM. Here, we attempt to directly minimize the overall probability of decoding failure PT{SM < A ( M ) } . First, we recast this optimization problem into a geometrical framework. Using this framework, we derive a simple modification to the KV algorithm which results in a provably better multiplicity assignment. Alternatively, we approximate the distribution of SM by a Gaussian distribution, and develop an iterative algorithm to minimize P r { s M 5 A ( M ) } under this approximation. This leads to coding gains of about 0.20dB for RS codes of length 255 and up to 0.75dB for RS codes of length 15, as compared to the Koetter-Vardy algorithm.

Iterative algebraic soft-decision list decoding of Reed-Solomon codes

“…Other algorithms of [10] and [11] minimize the error probability directly. The algorithm of [10] (Gauss) assumes a Gaussian distribution of the score, while that of [11] (Chernoff) minimizes a Chernoff bound on the error probability. The later appears to have the best performance.…”

“…These permutations may also be required for the JN algorithm. Then, the relative reduction in complexity is (10) For example, if we assume that on average , a simple calculation for the code over shows that the relative reduction in the complexity of the GE step is about 75%. In practice is close to one.…”

“…The performance of different ASD algorithms are compared for infinite interpolation costs, the KV algorithm [9], the Gaussian approximation (Gauss) [10], and the Chernoff bound algorithm (Chernoff) [11]. It is noted that the Chernoff bound algorithm has the best performance, especially at the tail of error probability.…”

“…Koetter and Vardy (KV) [9] developed an algebraic soft-decision decoding (ASD) algorithm for RS codes based on a multiplicity assignment scheme for the GS algorithm. Alternative ASD algorithms, such as the Gaussian approximation algorithm by Parvaresh and Vardy [10] and the algorithm by El-Khamy and McEliece based on the Chernoff bound [11], [12], have better performance. Jiang and Narayanan (JN) developed an iterative algorithm based on belief propagation for soft decoding of RS codes [13], [14].…”

Algebraic Soft-Decision Decoding of Reed-Solomon Codes with Erasures on Gaussian Channels

“…They also showed that for codes of infinite length the best way to choose the value of m is by making it proportional to the soft information available for each point. In [6], [7] and [8] there were presented some multiplicity assignment strategies which were optimized for infinite-length codes through complex numeric algorithms. However for codes of finite length, an optimum multiplicity assignment strategy is still an open problem.…”