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Wireless Communications and Mobile Computing

Construction of Quasi‐Cyclic LDPC Codes Based on Fundamental Theorem of Arithmetic

Hai Jiang Zhu, Liqun Pu, Hengzhou Xu, Bo Zhang

Published Jan 1, 2018• 15 citations
Publisher

Abstract

Quasi‐cyclic (QC) LDPC codes play an important role in 5G communications and have been chosen as the standard codes for 5Genhanced mobile broadband(eMBB) data channel. In this paper, we study the construction of QC LDPC codes based on an arbitrary given expansion factor (or lifting degree). First, we analyze the cycle structure of QC LDPC codes and give the necessary and sufficient condition for the existence of short cycles. Based on the fundamental theorem of arithmetic in number theory, we divide the integer factorization into three cases and present three classes of QC LDPC codes accordingly. Furthermore, a general construction method of QC LDPC codes with girth of at least 6 is proposed. Numerical results show that the constructed QC LDPC codes perform well over the AWGN channel when decoded with the iterative algorithms.

DOI
10.1155/2018/5264724

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Quasi‐cyclic (QC) LDPC codes play an important role in 5G communications and have been chosen as the standard codes for 5Genhanced mobile broadband(eMBB) data channel. In this paper, we study the construction of QC LDPC codes based on an arbitrary given expansion factor (or lifting degree). First, we analyze the cycle structure of QC LDPC codes and give the necessary and sufficient condition for the existence of short cycles. Based on the fundamental theorem of arithmetic in number theory, we divide the integer factorization into three cases and present three classes of QC LDPC codes accordingly. Furthermore, a general construction method of QC LDPC codes with girth of at least 6 is proposed. Numerical results show that the constructed QC LDPC codes perform well over the AWGN channel when decoded with the iterative algorithms.
Quasi-cyclic LDPC (QC-LDPC) codes have been accepted as the standard codes of 5G enhanced mobile broadband data channel. These standard codes are designed to support multiple lifting sizes and possess rate-compatible property, which can help adapt various information lengths and code rates well. In this paper, we propose analgebra-assistedmethod for constructing QC-LDPC codes with such properties. We will first review the encoding mechanism and requirements of 5G LDPC codes, and presentcycle analysisfor such emerging codes. We then propose a metric, referred to asweighted average number of cycles(WANC), from the perspective of cycle structure for constructing the QC-LDPC codes that can supportmultiple lifting sizes. Based on the WANC metric and algebraic methods, we develop a simple and practical algorithm to construct this kind of QC-LDPC codes. We finally apply the proposed algorithm to construct the exponent matrices for cases of 5G LDPC codes and the standard LDPC codes of consultative committee for space data systems, respectively. Simulation results show that the proposed WANC metric and designed algorithm are feasible and effective, and thus can be utilized to design other similar QC-LDPC codes.
In this letter, we present an efficient method for constructing a family of quasi-cyclic low-density parity-check (QC-LDPC) codes specified only by one single exponent matrix that can support multiple lifting sizes. Our method adopts the small-lifting-size-first design strategy and the resulting exponent matrix of larger lifting size has non-decreasing girth. Numerical results show that all constituent codes can perform as well as (outperform) the CCSDS (5G) LDPC codes over the AWGN channel with the iterative decoding algorithms. This fact demonstrates the feasibility and validity of our proposed method and makes it competitive in some applications, e.g., enhancing short and medium-length LDPC codes to support longer length.
This letter presents a necessary and sufficient condition for a class of quasi-cyclic low-density parity-check (QC LDPC) codes without girth four. Girth-four property of a class of QC LDPC codes is investigated. Good QC LDPC codes without girth four can be constructed by selecting proper shifting factors according to the proposed theorems. Examples are provided to verify the theorems. The simulation results show that the QC LDPC codes without girth four achieve a better BER performance compared with that of randomly constructed LDPC codes.
Designs and constructions of quasi-cyclic (QC) LDPC codes for the AWGN channel are presented. The codes are constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes and are referred to as RS-QC-LDPC codes. Several classes of RS-QC-LDPC codes are given. Cycle structural properties of the Tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented. The designed codes perform well in both waterfall and low error-rate regions.
In wireless communication, transmission error can be caused by multi-path fading, shadowing, scattering, interference and noise. As a result, errors generated in digital communication can be classified as burst errors, random bit errors, crosstalk and echo. For reliable communication, channel coding is used which adds redundancy to the information bits. Reliable and low latency communication are crucial for future communication systems involving satellite constellations and 5G communication systems, which underscores the importance of channel coding. Various codes exist for channel coding, among which low-density parity-check (LDPC) codes, polar codes and tail biting convolution codes are mentioned for 5G ultra-reliable low latency communication and enhanced mobile broadband systems. In the literature, the performance of large block-length codes has been well-studied. But these codes lead to high latency. The design of short block-length codes remains an open problem due to the trade-off between latency and performance. In this paper, we develop a systematic procedure for construction of hybrid quasi-cyclic LDPC (QC-LDPC) codes and compare the performance of the proposed approach with random LDPC and existing QC-LDPC codes for short block-lengths and phase shift keying (PSK) signal characteristics. The proposed method exploits Euclidean geometry and circulant decomposition resulting in a QC-LDPC matrix that avoids shorter girths for iterative decoding cycles. We emphasize that the proposed approach is different from the protographic construction method, masking method, identity matrix replacement method, algebraic based construction method, and base graph method used for the construction of QC-LDPC codes.
There are many algebraic tools, e.g., finite fields and combinatorial designs, for constructing good quasi-cyclic LDPC (QC-LDPC) codes. In particular, QC-LDPC codes designed based on two arbitrary subsets of a given finite field perform very well and some well-known algebraic constructions of QC-LDPC codes have been verified as special cases of this method. How to choose these two subsets to further improve the performance of finite field based QC-LDPC codes is of interest. In this paper, we study such codes and analyze their structural properties from isomorphism perspective, and generalize some rules to efficiently determine the non-isomorphic finite field based codes. By comparing the cycle distributions of non-isomorphic codes, we can easily construct the finite field based QC-LDPC codes with larger girth and fewer short cycles. These codes generally perform very well and can be used as the ingredients of some further processings, e.g., masking. Numerical results show that the constructed codes have better performance with the iterative decoding algorithms.