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IEEE International Conference on Communication Technology (ICCT)

An Algebraic Construction of Quasi-Cyclic LDPC Codes Based on the Conjugates of Primitive Elements over Finite Fields

Muhammad Asif, Wuyang Zhou, Juma Saidi Ally, Nauman Ali Khan, Zain Ul Abiden Akhtar

Published Oct 1, 2018• 3 citations
Publisher

Abstract

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.

DOI
10.1109/icct.2018.8600012

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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.
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.
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.
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.
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 the 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 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 this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC LDPC) codes based on cyclic subgroup over GF(q) . LDPC codes constructed by this method are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN) with sum product algorithm (SPA). The main advantage is that QC LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.
This correspondence presents a construction of quasicyclic (QC) low-density parity-check (LDPC) codes based on a special type of combinatorial designs known as block disjoint difference families (BDDFs). The proposed construction of QC-LDPC codes gives parity-check matrices with column weight three and Tanner graphs having a girth lower-bounded by 6. The proposed QC-LDPC codes provide an excellent performance with iterative decoding over an additive white Gaussian-noise (AWGN) channel. Performance analysis shows that the proposed short and moderate length QC-LDPC codes perform as well as their competitors in the lower signal-to-noise ratio (SNR) region but outperform in the higher SNR region. Also, the codes constructed are quasicyclic in nature, so the encoding can be done with simple shift-register circuits with linear complexity.