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IEEE Symposium on Computers and Communications (ISCC)

Construction of structured low density lattice codes based on finite fields

Jiayun Li, Shu‐Tao Xia, Xinji Liu

Published Jul 1, 2015• 0 citations
Publisher

Abstract

Finite fields were successfully used to construct algebraic low-density parity-check (LDPC) codes, especially Quasi-Cyclic LDPC codes. These LDPC codes with large minimum distances have lower error floor, linear complexity of encoding and are more practical for hard-decision algebraic decoding. In this paper, we show that finite fields can also be successfully used to construct algebraic low-density lattice codes (LDLC), denoted by structured LDLC. A general framework to construct algebraic LDLC is presented. LDLC constructed by this general framework have comparable performance to the corresponding random codes over addition white Gaussian noise (AWGN) channel with iterative soft-decision decoding in terms of symbol-error probability. Furthermore, the general framework is extended to complex low-density lattice codes (CLDLC) and results in algebraic CLDLC which perform very well for small dimensions.

DOI
10.1109/iscc.2015.7405587

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Finite fields were successfully used to construct algebraic low-density parity-check (LDPC) codes, especially Quasi-Cyclic LDPC codes. These LDPC codes with large minimum distances have lower error floor, linear complexity of encoding and are more practical for hard-decision algebraic decoding. In this paper, we show that finite fields can also be successfully used to construct algebraic low-density lattice codes (LDLC), denoted by structured LDLC. A general framework to construct algebraic LDLC is presented. LDLC constructed by this general framework have comparable performance to the corresponding random codes over addition white Gaussian noise (AWGN) channel with iterative soft-decision decoding in terms of symbol-error probability. Furthermore, the general framework is extended to complex low-density lattice codes (CLDLC) and results in algebraic CLDLC which perform very well for small dimensions.
In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.
Recently, there have been major developments in utilizing the finite fields to construct Low-density Parity-check (LDPC) codes. In this correspondence, an algebraic approach based on the conjugates of primitive elements over finite fields to construct Quasi-Cyclic (QC) Low-Density Parity-Check codes is presented. Proposed QC-LDPC codes provide an excellent error performance with Belief Propagation (BP) decoding over an Additive White Gaussian Noise (AWGN) channel. Based on numerical results, the performance analysis shows that the proposed QC-LDPC codes perform as well as the randomly constructed Progressive edge growth (PEG) LDPC codes and algebraic QC-LDPC in the lower signal-to-noise ratio (SNR) region but outperform their counterparts in the higher SNR region. Also, the codes constructed are QC in nature, so the encoding can be done with shift register circuits having linear complexity.
This paper presents a simple and very flexible method for constructing quasi-cyclic (QC) low density paritycheck (LDPC) codes based on finite fields. The code construction is based on two arbitrary subsets of elements from a given field. Some well known constructions of QC-LDPC codes based on finite fields and combinatorial designs are special cases of the proposed construction. The proposed construction in conjunction with a technique, known as masking, results in codes whose Tanner graphs have girth 8 or larger. Experimental results show that codes constructed using the proposed construction perform well and have low error-floors. Also presented in the paper is a reduced-complexity iterative decoding scheme for QC-LDPC codes based on the section-wise cyclic structure of their parity-check matrices. The proposed decoding scheme is an improvement of an earlier proposed reduced-complexity iterative decoding scheme.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes which are known as QC-LDPC codes, have many applications due to their simple encoding implementation by means of cyclic shift registers. In this paper, we construct QC-LDPC codes from group rings. A group ring is a free module (at the same time a ring) constructed in a natural way from any given ring and any given group. We present a structure based on the elements of a group ring for constructing QC-LDPC codes. Some of the previously addressed methods for constructing QC-LDPC codes based on finite fields are special cases of the proposed construction method. The constructed QC-LDPC codes perform very well over the additive white Gaussian noise (AWGN) channel with iterative decoding in terms of bit-error probability and block-error probability. Simulation results demonstrate that the proposed codes have competitive performance in comparison with the similar existing LDPC codes. Finally, we propose a new encoding method for the proposed group ring based QC-LDPC codes that can be implemented faster than the current encoding methods. The encoding complexity of the proposed method is analyzed mathematically, and indicates a significate reduction in the required number of operations, even when compared to the available efficient encoding methods that have linear time and space complexities.
In the late 1950's and early 1960's, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for correcting random errors with algebraic decoding, such as Bose-Chaudhuri-Hocqenghem (BCH) and Reed-Solomon (RS) codes. Recently it has been shown that finite fields can also be used successfully to construct binary quasi-cyclic (QC)-LDPC codes that perform very well not only over the AWGN channel but also over the binary erasure channel with iterative decoding, besides being efficiently encodable. This paper is concerned with constructions of nonbinary QC-LDPC codes based on finite fields
Multilevel LDPC Lattices With Efficient Encoding and Decoding and a Generalization of Construction D′\text{D}'D′
Lattice codes are elegant and powerful structures that not only achieve the capacity of the additive white Gaussian noise (AWGN) channel but are also a key ingredient to many multiterminal schemes that exploit linearity properties. However, constructing lattice codes that can realize these benefits with low complexity is still a challenging problem. In this paper, efficient encoding and decoding algorithms are proposed for multilevel binary low-density parity-check (LDPC) lattices constructed via Construction D' whose complexity is linear in the total number of coded bits. Moreover, a generalization of Construction D' is proposed, which relaxes some of the nesting constraints on the parity-check matrices of the component codes, leading to a simpler and improved design. Based on this construction, low-complexity multilevel LDPC lattices are designed whose performance under multistage decoding is comparable to that of polar lattices and close to that of low-density lattice codes on the power-unconstrained AWGN channel.