We present a detailed analysis of generalized minimum distance (GMD) decoding algorithms for Euclidean space codes. In particular, we completely characterize GMD decoding regions in terms of receiver front-end properties. This characterization is used to show that GMD decoding regions have intricate geometry. We prove that although these decoding regions are polyhedral, they are essentially always nonconvex. We furthermore show that conventional performance parameters, such as error-correction radius and effective error coefficient, do not capture the essential geometric features of a GMD decoding region, and thus do not provide a meaningful measure of performance. As an alternative, probabilistic estimates of, and upper bounds upon, the performance of GMD decoding are developed. Furthermore, extensive simulation results, for both low-dimensional and high-dimensional sphere-packings, are presented. These simulations show that multilevel codes in conjunction with multistage GMD decoding provide significant coding gains at a very low complexity. Simulated performance, in both cases, is in remarkably close agreement with our probabilistic approximations.

Generalized minimum-distance decoding in Euclidean-space: performance analysis

The statistical approach proposed by D. Agrawal and A. Vardy (see IEEE Trans. Inform. Theory, vol.46, pp.60-83, Jan. 2000) to evaluate the error performance of the generalized minimum distance (GMD) decoding is extended to other reliability based decoding algorithms for binary linear block codes, namely Chase-type, combined GMD and Chase-type, and ordered statistic decoding. In all cases, tighter and simpler bounds than previously proposed ones have been obtained with this approach.

Error performance analysis for reliability-based decoding algorithms

Order-w reprocessing is a suboptimal soft-decision decoding approach for binary linear block codes in which up to w bits are systematically flipped on the so-called most reliable (information) basis (MRB). This correspondence first incorporates two preprocessing rules into order-w reprocessing and shows that, with appropriate choice of parameters, the proposed order-w reprocessing with preprocessing requires comparable complexity to order-w reprocessing but achieves asymptotically the performance of order-(w+2) reprocessing. To complement the MRB, a second basis is employed for practical SNRs and this approach is systematically extended to a multibasis order-w reprocessing scheme for high signal-to-noise ratios (SNRs). It is shown that the proposed multibasis scheme significantly enlarges the error-correction radius, a commonly used measure of performance at high SNRs, over the original (single-basis) order-w reprocessing. As a by-product, this approach also precisely characterizes the asymptotic performance of the well-known Chase and generalized minimum distance (GMD) decoding algorithms. The proposed algorithm successfully decodes the (192,96) Reed-Solomon concatenated code and the (256,147) extended BCH code in near optimal manner (within 0.01 dB at a block-error rate of 10-5) with affordable computational cost

Soft-Decision Decoding of Linear Block Codes Using Preprocessing and Diversification

In this letter, Chase decoding algorithms are generalized into a family of bounded distance decoding algorithms, so that the conventional Chase algorithm-2 and Chase algorithm-3 become the two extremes of this family. Consequently, more flexibility in the tradeoffs between error performance and decoding complexity is provided by this generalization, especially for codes with large minimum distance. Finally this approach is extended to decoding with erasures.

Chase-type and GMD coset decodings

In this letter, the reliability-based decoding algorithms developed for binary linear codes are investigated for SDD of Reed-Solomon (RS) codes using their binary images. It is shown that if such approaches are promising, they also have to be carefully considered as their behavior at error rates which can be simulated often differ from that at lower error rates for which RS codes have been implemented in concatenated schemes.

Reliability-based decoding of Reed-Solomon codes using their binary image

This paper presents a unified method for evaluating the error-correction radii of many reliability-based soft-decision decoding algorithms for binary linear block codes. Based on this unified method, these decoding algorithms as well as their potential improvements can be compared directly. The error-correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft-decision decoder to the transmitted signal sequence. It is shown that this problem can be formulated as a constrained optimization problem. Although no general closed-form expression is possible, a simple algorithm that always converges to the optimum solution of this optimization problem is presented. Based on these results, the error-correction radii of some well-known reliability-based soft-decision decoding algorithms are revisited and compared.

Shu Lin

Papers

A unified method for evaluating the error-correction radius of reliability-based soft-decision algorithms for linear block codes