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IEEE International Symposium on Information Theory (ISIT)

Non-binary algebraic spatially-coupled quasi-cyclic LDPC codes

Keke Liu, Mostafa El‐Khamy, Jungwon Lee, Inyup Kang, Arvind Yedla

Published Jun 1, 2014• 5 citations
Publisher

Abstract

This paper considers the algebraic construction and performance of non-binary spatially-coupled low density parity check (LDPC) codes. A replicate-and-mask approach is presented to construct finite-length algebraic quasi-cyclic (QC) spatially-coupled (SC) LDPC codes. Numerical results show the superiority of non-binary algebraic SC QC LDPC codes over the corresponding random non-binary (block and SC) LDPC codes. In this paper, it is demonstrated that the threshold saturation phenomenon, previously demonstrated for binary SC LDPC codes, also holds for non-binary SC LDPC codes over the binary-input AWGN channel with BPSK modulation.

DOI
10.1109/isit.2014.6874885

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This paper considers the algebraic construction and performance of non-binary spatially-coupled low density parity check (LDPC) codes. A replicate-and-mask approach is presented to construct finite-length algebraic quasi-cyclic (QC) spatially-coupled (SC) LDPC codes. Numerical results show the superiority of non-binary algebraic SC QC LDPC codes over the corresponding random non-binary (block and SC) LDPC codes. In this paper, it is demonstrated that the threshold saturation phenomenon, previously demonstrated for binary SC LDPC codes, also holds for non-binary SC LDPC codes over the binary-input AWGN channel with BPSK modulation.
The replicate-and-mask (R&M) construction of finite-length spatially-coupled (SC) LDPC codes is proposed in this paper. The proposed R&M construction generalizes the conventional matrix unwrapping construction and contains it as a special case. The R&M construction of a class of algebraic spatially coupled (SC) quasi-cyclic (QC) LDPC codes over arbitrary finite fields is demonstrated. The girth, rank, and time-varying periodicity of the proposed R&M SC QC LDPC codes are analyzed. The error rate performance of finite-length nonbinary algebraic SC QC LDPC codes is investigated with window decoding. Compared to the conventional unwrapping construction, it is found through numerical simulations that the R&M construction resulted in SC QC LDPC codes with better block error rate performance and lower error floors. With a flooding schedule decoder, it is shown that the proposed R&M algebraic SC QC LDPC codes have better error performance than the corresponding LDPC block codes and random SC codes. The R&M construction of irregular SC QC LDPC codes is demonstrated. It is shown that low-complexity regular puncturing schemes can be deployed on these codes to construct families of rate-compatible irregular SC QC LDPC codes with good performance.
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.
We analyze the asymptotic performance of non-binary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group, when the transmission takes place over the binary erasure channel. We propose an efficient method to derive an upper bound to the maximum a posteriori probability (MAP) threshold for nonbinary LDPC codes, and observe that the MAP performance of regular LDPC codes improves with the alphabet size. We then consider nonbinary SC-LDPC codes. We show that the same threshold saturation effect experienced by binary SC-LDPC codes occurs for the nonbinary codes, hence we conjecture that the BP threshold for large termination length approaches the MAP threshold of the underlying regular ensemble.
In this paper, we deal with time-varing quasi-cyclic (QC) spatially-coupled (SC) low-density parity-check (LDPC) codes by lifting the coupled protographs. The construction method of girth-6 protographs for any code rate is proposed. We provide a theoretical lower bound on the syndrome former constraint length under condition of the protographs without 4-cycles in their Tanner graphs. Then we use periodic QC lifting method to search the QC-SC-LDPC matrices with larger girth based on the protographs. The minimum dimension size of QC-SC-LDPC codes is given to count the girth of QC-SC-LDPC codes with a semi- infinite binary-check matrix structure. The performance of such constructed codes with lower complexity and delay is superior to the LDPC block codes.
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.
On dispersing different base matrices combined with masking, different kinds of quasi-cyclic low-density parity check (QC-LDPC) codes are constructed, which may achieve good error performances. Two new algebraic constructions of regular QC-LDPC codes are presented in this paper. The two constructions are based on two base matrices by dispersing and masking. The two base matrices are based on the generators of the cyclic group of a prime field. Results show that the two classes of codes presented in this paper perform well with iterative decoding over AWGN channel and have advantages over MacKay codes in some aspects of code performances. We find that one of the constructed codes converges very fast with iterative decoding, which is a critical property in ultra-high throughput communication systems.