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arXiv:1201.3982

Min-Sum algorithm for lattices constructed by Construction D

Mehri, Hassan

Published Jan 19, 2012• 0 citations
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Abstract

The so-called min-sum algorithm has been applied for decoding lattices constructed by Construction D'. We generalize this iterative decoding algorithm to decode lattices constructed by Construction D. An upper bound on the decoding complexity per iteration, in terms of coding gain, label group sizes of the lattice and other factors is derived. We show that iterative decoding of LDGM lattices has a reasonably low complexity such that lattices with dimensions of a few thousands can be easily decoded.

DOI
10.48550/arxiv.1201.3982

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The so-called min-sum algorithm has been applied for decoding lattices constructed by Construction D'. We generalize this iterative decoding algorithm to decode lattices constructed by Construction D. An upper bound on the decoding complexity per iteration, in terms of coding gain, label group sizes of the lattice and other factors is derived. We show that iterative decoding of LDGM lattices has a reasonably low complexity such that lattices with dimensions of a few thousands can be easily decoded.
The so-called min-sum algorithm for iterative decoding of low-density parity-check (LDPC) codes is generalized to decode lattices. An upper bound on the decoding complexity per iteration is derived, and for LDPC lattices constructed by Construction D' and using a nested sequence of LDPC codes, exact values for computational complexity are also given. We show that iterative decoding of LDPC lattices has a reasonably low complexity such that lattices with dimensions of a few thousands can be easily decoded.
Low-density parity-check codes (LDPC) can have an impressive performance under iterative decoding algorithms. In this paper we introduce a method to construct high coding gain lattices with low decoding complexity based on LDPC codes. To construct such lattices we apply Construction D', due to Bos, Conway, and Sloane, to a set of parity checks defining a family of nested LDPC codes. For the decoding algorithm, we generalize the application of max-sum algorithm to the Tanner graph of lattices. Bounds on the decoding complexity are derived and our analysis shows that using LDPC codes results in low decoding complexity for the proposed lattices. The progressive edge growth (PEG) algorithm is then extended to construct a class of nested regular LDPC codes which are in turn used to generate low density parity check lattices. Using this approach, a class of two-level lattices is constructed. The performance of this class improves when the dimension increases and is within 3 dB of the Shannon limit for error probabilities of about 10-6. This is while the decoding complexity is still quite manageable even for dimensions of a few thousands
Iterative decoding of non-binary LDPC codes is currently performed using either the sum-product or the min-sum algorithms or slightly different versions of them. In this paper, several low-complexity quasi-optimal iterative algorithms are proposed for decoding non-binary codes. The min-max algorithm is one of them and it has the benefit of two possible LLR domain implementations: a standard implementation, whose complexity scales as the square of the Galois field's cardinality and a reduced complexity implementation called selective implementation, which makes the min-max decoding very attractive for practical purposes.
Low-density parity-check (LDPC) codes can have a good performance under iterative decoding algorithms. This idea is used to construct a class of lattices with relatively high coding gain and low decoding complexity. To construct such lattices, Construction D' is applied to the set of parity check vectors of a class of nested LDPC codes. Bounds on the minimum distance and the coding gain of the corresponding lattice are provided. We also provide a practical way of finding the cross sections of the lattice given the parity check matrix of a lattice. The progressive edge growth algorithm is extended and is used to construct a class of nested binary codes to generate the corresponding lattice. Simulation results confirm the good performance of this class of lattice.