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Lecture notes in computer science

On Linear Programming Decoding on a Quantized Additive White Gaussian Noise Channel

Eirik Rosnes

Published Jan 1, 2009• 0 citations
Publisher
DOI
10.1007/978-3-642-10868-6_2

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We consider data transmission over a binary-input additive white Gaussian noise channel using low-density parity-check codes. One of the most popular techniques for decoding low-density parity-check codes is the linear programming decoder. In general, the linear programming decoder is suboptimal. In this paper we present a systematic approach to enhance the linear programming decoder. More precisely, in the cases where the linear program outputs a fractional solution, we give a simple algorithm to identify frustrated cycles which cause the output of the linear program to be fractional. Then adding these cycles, adaptively to the basic linear program, we show improved word error rate performance.
We consider transmission over a binary-input additive white Gaussian noise channel using low-density parity-check codes. One of the most popular techniques for decoding low-density parity-check codes is the linear programming decoder. In general, the linear programming decoder is suboptimal. I.e., the word error rate is higher than the optimal, maximum a posteriori decoder. In this paper we present a systematic approach to enhance the linear program decoder. More precisely, in the cases where the linear program outputs a fractional solution, we give a simple algorithm to identify frustrated cycles which cause the output of the linear program to be fractional. Then adding these cycles, adaptively to the basic linear program, we show improved word error rate performance.
A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates LP-based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ldquomaximum-likelihood (ML) certificaterdquo property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with LP decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for ternary Golay code with ternary phase-shift keying (PSK) modulation over additive white Gaussian noise (AWGN), and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision-based decoding. LP decoding is also simulated for medium-length ternary and quaternary low-density parity-check (LDPC) codes with corresponding PSK modulations over AWGN.