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International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON)

A high rate non-binary QC-LDPC codes in high order Galois Field for PR2 and EPR2 channels

Mongkol Kupimai

Published May 1, 2014• 0 citations
Publisher

Abstract

In this paper, the new structure of high rate non-binary QC-LDPC codes based on the circulant permutation matrices is proposed. The construction of parity-check matrix of the proposed codes is obtained from (2, L) regular QC-LDPC codes. The parity-check matrix of the proposed codes has girth greater than 8. Simulation results over PR channels with the high order Galois Field show that the bit error rate and the symbol error rate performance of the proposed codes is better than those of the random non-binary LDPC codes, both in PR2 channel and EPR2 channel.

DOI
10.1109/ecticon.2014.6839791

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In this paper, the new structure of high rate non-binary QC-LDPC codes based on the circulant permutation matrices is proposed. The construction of parity-check matrix of the proposed codes is obtained from (2, L) regular QC-LDPC codes. The parity-check matrix of the proposed codes has girth greater than 8. Simulation results over PR channels with the high order Galois Field show that the bit error rate and the symbol error rate performance of the proposed codes is better than those of the random non-binary LDPC codes, both in PR2 channel and EPR2 channel.
ABSTRACT In this paper, we propose a modified high rate (3,L) quasi‐cyclic low‐density parity‐check (QC‐LDPC) code based on circulant permutation matrices for efficient encoder implementation. The structure of the code is modified from a (3,L) regular QC‐LDPC code to be an approximate lower triangular matrix and a linear independent form. We present two novel efficient encoding techniques for generating redundant parity bits. The complexity of the encoder implementation depends on the number of redundant parity bits of the code for one‐stage encoding and the length of the code for two‐stage encoding. The advantage of both encoding techniques is that few XOR gates are used in encoder implementation. The simulation results show that the proposed codes at code rate 0.884 and 0.953 have good bit error rate performance compared with other LDPC codes over PR2 and EPR2 channels. European Transactions on Telecommunications. Copyright © 2012 John Wiley & Sons, Ltd.
In this paper, we propose new constructions for regular girth-8 quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices (CPM). The constructions assume symmetries in the structure of the parity-check matrix and employ a greedy exhaustive search algorithm to find the permutation shifts of the CPMs. As a result of symmetries, the new codes have a more compact representation compared with their counterparts. In majority of cases, also, they achieve the girth 8 at a shorter block length for the same degree distribution (code rate). Deterministic (explicit) constructions are also presented to expand the proposed parity-check matrices to larger block lengths and higher rates. The proposed long high-rate codes are often substantially shorter than regular girth-8 QC-LDPC codes of similar rate in the literature. Simulation results demonstrate that the proposed symmetric codes have competitive performance in comparison with similar existing QC-LDPC codes that lack symmetry.
The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10-12. Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.
Cyclically-coupled quasi-cyclic low-density parity-check (CC-QC-LDPC) codes are a new class of QC-LDPC codes which can achieve excellent error performance and relatively low hardware requirement. In this paper, we modify the CC-QC-LDPC code construction by using random permutation matrices instead of circulant permutation matrices, forming the “random-permutation-matrix-based CC-LDPC (RP-CC-LDPC) code”. The objective is to achieve a better error performance under the same code length. We simulate the bit error rate using FPGA simulations. We also compare the BER results and decoder complexity of the above codes with those of regular and irregular QC-LDPC codes under the same code length and code rate.
For a high-rate case, it is difficult to randomly construct good low-density parity-check (LDPC) codes of short and moderate lengths because their Tanner graphs are prone to have short cycles. Also, the existing high-rate quasi-cyclic (QC) LDPC codes can be constructed only for very restricted code parameters. In this paper, based on special classes of cyclic difference families, we propose a new construction method of high-rate regular QC LDPC codes having parity-check matrices consisting of a single row of circulants with column-weight 3 or 4. The proposed QC LDPC codes can be constructed for various code rates and lengths including the minimum achievable length for given column-weight and design rate under girth 6. It is observed that the parity-check matrices of the proposed QC LDPC codes have full rank for column-weight 3 and just one redundant row for column-weight 4. It is shown that the error correcting performance of the proposed QC LDPC codes of short and moderate lengths is almost the same as that of the existing ones through numerical analysis.