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International Conference on Telecommunications and Signal Processing (TSP)

Column weights optimization for semi-regular nonbinary LDPC codes

Wojciech Sułek, Marcin Kucharczyk

Published Jul 1, 2015• 1 citation
Publisher

Abstract

The paper concerns Low Density Parity Check (LDPC) codes over nonbinary Galois Fields GF(q), that are a generalization of the industrial standard binary LDPC codes for forward error correction in communication and information systems. The semi-regular nonbinary LDPC codes can outperform their binary counterparts in the case of short to moderate code word lengths. The methods for the good code construction based on a computer search techniques are already well investigated. However, there is a lack of systematic results for optimal weight distribution of columns of parity check matrices for codes over different field order and different rate. In this article we present a Monte Carlo like algorithm for column weight optimization devoted to nonbinary semi-regular codes along with optimization results that facilitate the codes construction computer-based methods.

DOI
10.1109/tsp.2015.7296246

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The paper concerns Low Density Parity Check (LDPC) codes over nonbinary Galois Fields GF(q), that are a generalization of the industrial standard binary LDPC codes for forward error correction in communication and information systems. The semi-regular nonbinary LDPC codes can outperform their binary counterparts in the case of short to moderate code word lengths. The methods for the good code construction based on a computer search techniques are already well investigated. However, there is a lack of systematic results for optimal weight distribution of columns of parity check matrices for codes over different field order and different rate. In this article we present a Monte Carlo like algorithm for column weight optimization devoted to nonbinary semi-regular codes along with optimization results that facilitate the codes construction computer-based methods.
Low Density Parity Check (LDPC) codes over nonbinary Galois Fields GF(q) are a generalization of the industrial standard binary LDPC codes for forward error correction in communication and information systems.The nonbinary codes can achieve significantly better performance for short and moderate block lengths.A lot of works concerning "good" LDPC codes parity check matrix construction has been published so far.However, it is well known that efficient partially parallel hardware decoder architectures are allowed only for codes with blockwise partitioned structure of the parity check matrix, called structured codes.In this paper we present a versatile algorithm for construction of codes that are both nonbinary and structured.The proposed algorithm aims at optimizing the code graph (Tanner graph) by reducing the existence of small cycles with low external connectivity, while at the same time selecting appropriate nonzero coefficients from the Galois Field under interest.The algorithm can be used for code construction of any field order, block length and code rate.
In this paper, we focus on the design of a type of low- density parity-check (LDPC) codes called cycle codes whose parity-check matrix have exactly weight-2 columns. An approach to design nonbinary LDPC cycle codes with good performance is presented based on cycle entropy defined here. This method can obtain an elegant distribution of the nonbinary elements over the Galois Field GF(q) among the cycles whose length is exactly equal to the girth. Simulation results demonstrate that the LDPC codes designed with the proposed algorithm perform better than the randomly-constructed nonbinary LDPC cycle codes.
Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.
In this paper, we focus on short-block nonbinary LDPC (NB-LDPC) codes based on cyclic codes. Based on Tanner graphs' isomorphism, we present an efficient search algorithm for finding non-isomorphic binary cyclic LDPC codes. Notice that the parity-check matrix H of the resulting code is square and not of full rank, and its row weight and column weight are the same. By replacing the ones in the same column of H with a nonzero element of finite fields GF(q), a class of NB-LDPC codes over GF(q) is obtained. Numerical results show that the constructed codes perform well over the AWGN channel and have fast decoding convergence. Therefore, the proposed NB-LDPC codes provide a promising coding scheme for low-latency and high-reliability communications.
We propose a novel approach for optimization of nonbinary (NB) quasi-cyclic (QC)-LDPC codes. In this approach, the base parity-check matrices are constructed by the simulated annealing method, and then labeled while maximizing the so-called generalized girth of the NB LDPC code Tanner graph. Random coding bounds on the ML decoding error probability for ensembles of "almost regular" NB LDPC codes of finite lengths over extensions of the binary Galois field are derived. These bounds are based on the average bit weight spectra for the ensembles of NB LDPC codes. The observed FER performance of the sum-product BP decoding of "almost regular" NB QC-LDPC block codes is presented and compared to the finite-length random coding bounds, as well as to the performance of the optimized binary QC-LDPC block code in the 5G standard. In the waterfall region, the gap between the finite-length bounds on the error probability of the ML decoding and the simulation performance of the BP decoding is about 0.1 – 0.2 dB.
A unified approach to search for and optimize codes determined by their sparse parity-check matrices is presented. Replacing the nonzero elements of a binary parity-check matrix (the base or parent matrix) either by circulants or by companion matrices of elements from a finite field GF(2m), we obtain quasi-cyclic low-density parity-check (LDPC) block codes and binary images of nonbinary LDPC block codes, respectively. By substituting monomials of a formal variable D, we obtain the polynomial description of an LDPC convolutional code. A set of performance measures applicable to different classes of LDPC codes is considered, and a greedy algorithm for code performance optimization is presented. The heart of the new optimization algorithm is a fast procedure for searching for LDPC codes with large girth of their Tanner graphs. For a few classes of LDPC codes, examples of codes combining good error-correcting performance with compact representation are obtained. In particular, we present optimized convolutional LDPC codes and conclude that the LDPC block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered. Moreover, a specific channel model can easily be embedded into the optimization loop. Thereby, the code can be optimized for a specific channel. The efficiency of such an optimization is demonstrated via an example of faster than Nyquist (FTN) signaling using LDPC codes. The FTN strategy combined with a rate R = 1/2 LDPC code of length 64800 optimized for effective data rate R = 3/4 gains more than 0.5 dB compared with the standard LDPC codes of the same rate and length. The obtained gain corresponds to transmission at the capacity of the binary input additive white Gaussian noise channel. In most numerical examples, we consider codes with bidiagonal structure of the parity-check matrix. This restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards.