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Analysis of error floors for non-binary LDPC codes over general linear group through q-ary memoryless symmetric channels

隆之 野﨑, Kenta Kasai, KOHICHI SAKANIWA

Published Dec 1, 2012• 0 citations
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Abstract

In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over general linear groups with those for non-binary LDPC codes over finite fields transmitted through the q-ary memoryless symmetric channels under belief propagation decoding.To analyze non-binary LDPC codes defined over both the general linear group GL(m, F 2 ) and the finite field F 2 m , we investigate non-binary LDPC codes defined over GL(m 3 , F 2 m 4 ).We propose a method to lower the error floors for non-binary LDPC codes.In this analysis, we see that the non-binary LDPC codes constructed by our proposed method defined over general linear group have the same decoding performance in the error floors as those defined over finite field.The non-binary LDPC codes defined over general linear group have more choices of the labels on the edges which satisfy the condition for the optimization.

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In this paper, we investigate the error floors of the non-binary low-density parity-check codes transmitted over the binary erasure channels under belief propagation decoding. We propose a method to improve the decoding erasure rates in the error floors by optimizing labels in zigzag cycles in the Tanner graphs of codes. Furthermore, we give lower bounds on the bit and the symbol erasure rates in the error floors. The simulation results show that the presented lower bounds are tight for the codes designed by the proposed method.

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences•Jan 1, 2012•3 citations
In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over general linear groups with those for non-binary LDPC codes over finite fields transmitted through the q-ary memoryless symmetric channels under belief propagation decoding.To analyze non-binary LDPC codes defined over both the general linear group GL(m, F 2 ) and the finite field F 2 m , we investigate non-binary LDPC codes defined over GL(m 3 , F 2 m 4 ).We propose a method to lower the error floors for non-binary LDPC codes.In this analysis, we see that the non-binary LDPC codes constructed by our proposed method defined over general linear group have the same decoding performance in the error floors as those defined over finite field.The non-binary LDPC codes defined over general linear group have more choices of the labels on the edges which satisfy the condition for the optimization.
In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over general linear groups with those for non-binary LDPC codes over finite fields transmitted through the q-ary memoryless symmetric channels under belief propagation decoding. To analyze non-binary LDPC codes defined over both the general linear group GL(m,F2)\mathrm{GL}(m,\mathbb{F}_2)GL(m,F2​) and the finite field F2m\mathbb{F}_{2^m}F2m​, we investigate non-binary LDPC codes defined over GL(m3,F2m4)\mathrm{GL}(m_3,\mathbb{F}_{2^{m_4}})GL(m3​,F2m4​​). We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the non-binary LDPC codes constructed by our proposed method defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels on the edges which satisfy the condition for the optimization.
In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over the general linear group with those for non-binary LDPC codes over finite field transmitted over the q-ary memoryless symmetric channel under belief propagation decoding. To analyze non-binary LDPC codes defined over both general linear group GL(m, F2) and finite field F2m, we investigate non-binary LDPC codes defined over GL(m3, F2m4). We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the optimized non-binary LDPC codes defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels in the edges which satisfy the condition for the optimization.
Low-density parity-check (LDPC) codes are linear codes defined by sparse parity check matrices.Due to the sparseness of the parity check matrices, LDPC codes are efficiently decoded by the belief propagation (BP) decoder.It is known that non-binary LDPC codes can outperform binary ones.However, the decoding complexity grows with the size of non-binary alphabets.Hence, there is a trade-off between performance and complexity.In this dissertation, we focus on both binary and non-binary LDPC codes.We analyze the decoding error probability for finite length LDPC codes under BP decoding.The curve of the decoding error probability for finite length LDPC codes, as a function of channel error probability, is divided into two regions called waterfall region and error floor region.We analyze the decoding error probability in the waterfall region for binary LDPC codes and in the error floor region for non-binary LDPC codes.The main results of this dissertation are summarized as follows:• We analyze the decoding erasure probabilities of waterfall regions for binary LDPC codes over binary erasure channels (BECs) without any assumptions by analytically solving the covariance evolution.• We propose a method to lower the decoding error rates in the error floors of non-binary LDPC codes defined over Galois field and general linear group transmitted over the BEC, the memoryless binary-input output-symmetric (MBIOS) channel and q-ary memoryless symmetric (q-MS) channel.Simulation results show that the decoding error rates of the codes designed by the proposed method outperform those of the codes designed by the conventional method proposed by Poulliat et al.• We give lower bounds of decoding error rates in the error floor regions for non-binary LDPC codes over the BEC, MBIOS channel and q-MS channel.Simulation results show that those lower bounds are tight.Moreover, we show that the decoding error rates for non-binary LDPC codes defined over general linear group have same decoding performance in the error floor regions with that for non-binary LDPC codes defined over Galois field.Furthermore, we show that this tight lower bound monotonically decreases, as the order of Galois field of non-binary LDPC code increases in the BEC and binary additive white Gaussian noise channel.iii • We derive the weight distribution of the decoding error patterns in the BP decoder for non-binary LDPC codes defined over general linear group transmitted over the BEC.
In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels. We provide a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the belief propagation decoder. We consider an expurgated ensemble of non-binary LDPC codes by using the above necessary and sufficient condition, and hence exhibit lower error floors. Finally, we show lower bounds of the error floors for the expurgated LDPC code ensembles over the MBIOS channels.
In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels.We provide a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the belief propagation decoder.We consider an expurgated ensemble of non-binary LDPC codes by using the above necessary and sufficient condition, and hence exhibit lower error floors.Finally, we show lower bounds of the error floors for the expurgated LDPC code ensembles over the MBIOS channels.
In this paper, we investigate the error floors of the non-binary low-density parity-check codes transmitted over the binary erasure channels under belief propagation decoding. We propose a method to improve the decoding erasure rates in the error floors by optimizing labels in zigzag cycles in the Tanner graphs of codes. Furthermore, we give lower bounds on the bit and the symbol erasure rates in the error floors. The simulation results show that the presented lower bounds are tight for the codes designed by the proposed method.