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IEEE International Symposium on Information Theory (ISIT)

Construction of Algebraic-Based Variable-Rate QC-LDPC Codes

Huaan Li, Baoming Bai, Hengzhou Xu, Chao Chen

Published Jul 12, 2021• 2 citations
Publisher

Abstract

In this paper, we concentrate on one algebraic-based quasi-cyclic low-density parity-check (QC-LDPC) code constructed from two subsets of a finite field and generalize it to propose a class of variable-rate QC-LDPC (VR-QC-LDPC) codes, whose parity-check matrices are nested horizontally and have constant number of rows. Thus the proposed codes are significant at least in terms of storage complexity and can be simply implemented. The constructed codes also inherit the original algebraic-based QC-LDPC codes and their exponent matrices can be obtained from two subsets of the given finite field. We hereby analyze the structural properties from the isomorphism perspective, and present some rules to significantly prune the size of search space and determine the non-isomorphic exponent matrices. By distinguishing the smaller quantities of non-isomorphic matrices with cycle property metric, we can easily construct a series of nested exponent matrices with better cycle distributions and obtain the VR-QC-LDPC codes. Numerical results demonstrate that the constructed codes have better iterative decoding performance within a range of code rates and decoding iterations.

DOI
10.1109/isit45174.2021.9518132

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In this paper, we concentrate on one algebraic-based quasi-cyclic low-density parity-check (QC-LDPC) code constructed from two subsets of a finite field and generalize it to propose a class of variable-rate QC-LDPC (VR-QC-LDPC) codes, whose parity-check matrices are nested horizontally and have constant number of rows. Thus the proposed codes are significant at least in terms of storage complexity and can be simply implemented. The constructed codes also inherit the original algebraic-based QC-LDPC codes and their exponent matrices can be obtained from two subsets of the given finite field. We hereby analyze the structural properties from the isomorphism perspective, and present some rules to significantly prune the size of search space and determine the non-isomorphic exponent matrices. By distinguishing the smaller quantities of non-isomorphic matrices with cycle property metric, we can easily construct a series of nested exponent matrices with better cycle distributions and obtain the VR-QC-LDPC codes. Numerical results demonstrate that the constructed codes have better iterative decoding performance within a range of code rates and decoding iterations.
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.
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.
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.
Cyclic and quasi-cyclic algebraic LDPC codes constructed based on finite fields, finite geometries, and combinatorial designs can achieve excellent performance in terms of error rate, error floor and rate of decoding convergence with iterative decoding. However, the relatively high density of the parity-check matrix of an algebraic cyclic or quasi-cyclic LDPC code makes the hardware implementation complexity of the decoder quite large, which may be a critical issue in practical applications. This paper presents an effective reduced-complexity algorithm for decoding algebraic cyclic and quasi-cyclic LDPC codes based on the block cyclic structure and cyclic grouping of the rows of their parity-check matrices. The decoding of a code is carried out based on a single small submatrix of the parity-check matrix of the code in a revolving manner. The proposed decoding algorithm significantly reduces the hardware implementation complexity and the size of memory required to store information.
This paper presents a reduced-complexity iterative decoding scheme for quasi-cyclic (QC) low-density parity-check (LDPC) codes which is devised based on the section-wise cyclic structure of their parity-check matrices. The decoding scheme significantly reduces the hardware implementation complexity of a QC-LDPC code decoder in terms of the number of message processing units and the number of wires required to connect the message processing units. A long high-rate high-performance QC-LDPC code is used to demonstrate the effectiveness of the proposed decoding scheme. Using this long code as the mother code, descendant QC-LDPC codes of various lengths and rates can be constructed by puncturing and masking the columns and rows of the parity-check matrix of the mother code. These descendant codes can be decoded by deactivating specific sets of message processing units of the mother decoder.