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clidean-Space e inimurn-Distance Decodi

G. David Forney

Published Jan 1, 1996• 0 citations

Abstract

It is shown that multistage generalized minimum- distance (GMD) decoding of Euclidean-space codes and lattices can provide an excellent tradeoff between performance and com- plexity. We introduce a reliability metric for Gaussian channels that is easily computed from an inner product, and prove that a multistage GMD decoder using this metric is a bounded- distance decoder up to the true packing radius. The effective error coefficient of multistage GMD decoding is determined. Two simple modifications in the GMD decoding algorithm that drastically reduce this error coefficient are proposed. It is shown that with these modifications GMD decoding achieves the error coefficient of maximum-likelihood decoding for block codes and for generalized Construction A lattices. Multistage GMD decod- ing of the lattices Dg, Ea, K12, BW16, and A24 is investigated in detail. For 1<12,BW~, and A24, the GMD decoders have considerably lower complexity than the best known maximum- likelihood or bounded-distance decoding algorithms, and appear to be the most practically attractive decoders available. For high-dimensional codes and lattices (2 64 dimensions) maximum- likelihood decoding becomes infeasible, while GMD decoding algorithms remain quite practical. As an example, we devise a multistage GMD decoder for a 128-dimensional sphere packing with a nominal coding gain of 8.98 dB that attains an effective error coefficient of 1 365 760. This decoder requires only about 400 real operations, in addition to algebraic errors-and-erasures decoding of certain BCH and Hamming codes. It therefore ap- pears to be practically feasible to implement algebraic multistage GMD decoders for high-dimensional sphere packings, and thus achieve high effective coding gains.

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Cambridge University Press eBooks•Feb 23, 2006•0 citations

It is shown that multistage generalized minimum- distance (GMD) decoding of Euclidean-space codes and lattices can provide an excellent tradeoff between performance and com- plexity. We introduce a reliability metric for Gaussian channels that is easily computed from an inner product, and prove that a multistage GMD decoder using this metric is a bounded- distance decoder up to the true packing radius. The effective error coefficient of multistage GMD decoding is determined. Two simple modifications in the GMD decoding algorithm that drastically reduce this error coefficient are proposed. It is shown that with these modifications GMD decoding achieves the error coefficient of maximum-likelihood decoding for block codes and for generalized Construction A lattices. Multistage GMD decod- ing of the lattices Dg, Ea, K12, BW16, and A24 is investigated in detail. For 1<12,BW~, and A24, the GMD decoders have considerably lower complexity than the best known maximum- likelihood or bounded-distance decoding algorithms, and appear to be the most practically attractive decoders available. For high-dimensional codes and lattices (2 64 dimensions) maximum- likelihood decoding becomes infeasible, while GMD decoding algorithms remain quite practical. As an example, we devise a multistage GMD decoder for a 128-dimensional sphere packing with a nominal coding gain of 8.98 dB that attains an effective error coefficient of 1 365 760. This decoder requires only about 400 real operations, in addition to algebraic errors-and-erasures decoding of certain BCH and Hamming codes. It therefore ap- pears to be practically feasible to implement algebraic multistage GMD decoders for high-dimensional sphere packings, and thus achieve high effective coding gains.

Detailed geometric analysis of decoding regions for GMD decoding is presented. We show that GMD decoding regions are non-convex in essentially all cases of interest. We also prove that these regions are always bounded by hyperplane segments. For d-erasure GMD decoding, we establish the presence of a large number of bounding hyperplanes at distance /spl les/d+1. These results invalidate the estimates of performance derived from the union bound in the case of (multistage) GMD decoding. Alternative probabilistic estimates of, and upper bounds upon, the performance of GMD decoding are developed. Simulation results, for both low-dimensional and high-dimensional sphere packings, are in remarkably close agreement with our probabilistic approximations.

The error-correction capability and the performance of generalized minimum distance decod- ing based on threshold erasing is studied as a function of the number of decoding trials and of the allowable reliability values. Generalized Minimum Distance (GMD) decoding, as intro- duced by Forney (l), is a multitrial decoding algorithm that combines reliability information, represented by numbers in the closed interval (0,1), with algebraic decoding algorithms for error correction. In each trial, a fixed number of the most unreliable symbols in the received sequence is erased. If a cer- tain distance criterion is satisfied, Forney has shown that the transmitted codeword can be retrieved in at most (d/2) trials, where d is the Hamming distance of the code. In this work, we study a variation on Forney's scheme where, in each trial, symbols whose reliabilities are below a certain threshold are erased, and hence there is no need to sort the received symbols according to reliability values be- fore erasing as in Forney's GMD decoding. Furthermore, we consider limiting the allowable reliability values to the set R, = u~=mo/2'{1 - 2i/m}, where m = 1,2,3,. . .. This covers the simple cases of taking only hard decisions (m = l), allow- ing erasures (m = 2), or accommodating the full range (0,1) of reliability values as in Forney's GMD decoding (m + 00). Limiting the reliability values may be of importance since, in any digital implementation, the reliability values need to be quantized. Moreover, in some applications, e.g., GMD decod- ing of concatenated codes over discrete channels, the reliabil- ity values are discrete in nature (2). The maximum number of trials is kept as a parameter. Limiting the reliability values and the number of trials leads to the concept of reduced GMD decoding.

A multistage decoding algorithm is given for lattices obtained from a multilevel code formula. The algorithm is shown to have the same effective error-correcting radius as maximum-likelihood decoding, so that the performance loss is essentially determined by the increase in the effective error coefficient, for which an expression is given. The code formula generalizes some previous multilevel constructions to constructions of known single-level binary lattices with many levels, and then to decoders for them with the proposed algorithm. The trade-off between complexity reduction and performance loss is evaluated for several known lattices and two new ones, indicating that the approach is effective provided the binary codes involved in the code formula are not too short. All codes used in our constructions are binary Reed-Muller codes.>

A multistage decoding algorithm is given for lattices obtained from a generalized code formula. The corresponding multilevel construction is based on a chain of two-way lattice partitions and a family of binary linear block codes, whereas the multistage algorithm involves the use of a maximum-likelihood (ML) decoding algorithm for each two-way lattice partitions as well as a soft-decision ML decoding algorithm for each binary linear block code. The algorithm is shown to have the same effective error-correcting radius as ML decoding. Several known lattices and two new ones were then constructed by the generalized code formula. The trade-off between complexity reduction and performance loss achieved by the proposed algorithm is presented.

In this chapter, we continue the discussion on concatenated codes, which was initiated in Section 5.4. The main message to be conveyed in this chapter is that by using concatenation, one can obtain codes with favorable asymptotic performance—in a sense to be quantified more precisely—while the complexity of constructing these codes and decoding them grows polynomially with the code length. We first present a decoding algorithm for concatenated codes, due to Forney. This algorithm, referred to as a generalized minimum distance (in short, GMD) decoder, corrects any error pattern whose Hamming weight is less than half the product of the minimum distances of the inner and outer codes (we recall that this product is a lower bound on the minimum distance of the respective concatenated code). A GMD decoder consists of a nearest-codeword decoder for the inner code, and a combined error–erasure decoder for the outer code. It then enumerates over a threshold value, marking the output of the inner decoder as erasure if that decoder returns an inner codeword whose Hamming distance from the respective received sub-word equals or exceeds that threshold. We show that under our assumption on the overall Hamming weight of the error word, there is at least one threshold for which the outer decoder recovers the correct codeword. If the outer code is taken as a GRS code, then a GMD decoder has an implementation with time complexity that is at most quadratic in the length of the concatenated code.