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International Symposium on Turbo Codes and Iterative Information Processing

Construction of non-binary LDPC codes: A matrix-theoretic approach

Juane Li, Keke Liu, Shu Lin, Khaled Abdel-Ghaffar

Published Aug 1, 2014• 2 citations
Publisher

Abstract

This paper presents a matrix-theoretic approach to the construction of non-binary LDPC codes. Two algebraic methods for constructing non-binary LDPC codes are presented. The proposed construction methods have several ingredients including base matrix, matrix-dispersion, masking and replacement. By proper choice and combination of these ingredients, non-binary LDPC codes with good performance can be constructed.

DOI
10.1109/istc.2014.6955115

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This paper presents a matrix-theoretic approach to the construction of non-binary LDPC codes. Two algebraic methods for constructing non-binary LDPC codes are presented. The proposed construction methods have several ingredients including base matrix, matrix-dispersion, masking and replacement. By proper choice and combination of these ingredients, non-binary LDPC codes with good performance can be constructed.
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This paper unveils a strong connection between two major constructions of LDPC codes, namely the algebraic-based and the protograph-based constructions. It is shown that, from a graph-theoretic point of view, an algebraic LDPC code whose parity-check matrix is an array of submatrices of the same size over a finite field is a protograph LDPC code. Conversely, from a matrix-theoretic point of view, since the parity-check matrix of a protograph code can be arranged as an array of submatrices of the same size over a finite field and its base graph (or base matrix) can be constructed algebraically, a protograph LDPC code is an algebraic LDPC code. These two major approaches have their advantages and disadvantages in code construction. Unification of these two approaches may lead to better designs and constructions of LDPC codes to achieve good overall performance in terms of error performance in waterfall region, error-floor location and rate of decoding convergence. This paper is the first part of a series of two parts, Part-I and Part-II. Part-I investigates only the binary LDPC codes constructed by the superposition and the protograph-based methods. Part-II explores nonbinary LDPC codes from both superposition and protograph points of view. Also included in Part II are specific superposition constructions of both binary and nonbinary quasi-cyclic LDPC codes.
An algorithm for constructing parity-check matrices of non-binary quasi-cyclic low-density parity-check (NB QC-LDPC) codes is proposed. The algorithm finds short cycles in the base matrix and tries to eliminate them by selecting the circulants and the elements of GF(q). Firstly the algorithm tries to eliminate the cycles with the smallest number edges going outside the cycle. The efficiency of the algorithm is demonstrated by means of simulations. In particular, it was shown that NB QC-LDPC codes constructed with use of our algorithm loose less that 0.1 dB in comparison to the best NB LDPC codes.
This paper presents an original method for constructing high performance non-binary LDPC codes based on algebraic geometric codes’ parity check matrix. Codes constructed based on this method perform very well over the AWGN channel. With iterative decoding using a Fast Fourier Transform (FFT) based q-ary sum-product algorithm (QSPA), AG-LDPC codes achieve significantly large coding gains over the PEG non-binary LDPC and binary LDPC. Simulation results also show that the AG-LDPC significantly lower the decoding error floor, and the undetected error rate.
A new combinatorial technique for constructing girth-6 structured binary regular low-density parity-check (LDPC) codes based on special types of t-designs is given. A very large number of well-known t-designs can be used by this method for code construction. Based on this method, a t-(v,k,λ) design D=(X,B) can be exploited for code construction if it satisfies the following three conditions: 1) |B1 ∩ B2|≤ t for any two blocks B1,B2∈ B and B1≠ B2; 2) λ>1; and 3) k>t. Though the technique works for any t-design satisfying these conditions, we focus only on the utilization of simple triple systems, super-simple BIBDs, Steiner systems, and large sets (LSs) of t-designs. We also construct binary and non-binary girth-6 QC-LDPC codes from the t-designs satisfying these conditions by using matrix dispersion method. Experimental results show that the constructed non-binary QC-LDPC codes can provide good practical performance under iterative decoding using the fast Fourier transform based q-ary sum-product algorithm (FFT-QSPA) and they can achieve acceptable coding gains over random-like codes of comparable parameters decoded with sum-product algorithm (SPA).