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IEICE Technical Report; IEICE Tech. Rep.

Bounds of MAP Decoding Performance for Non-binary LDPC Codes

Eri Mutaguchi, Kasai Kenta, Tomoharu Shibuya, Kohichi Sakaniwa

Published Sep 4, 2008• 0 citations
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Related papers

Design and decoding algorithms of non-binary LDPC codes

Kenta Kasai, Kohichi Sakaniwa

IEICE Technical Report; IEICE Tech. Rep.•Dec 3, 2009•0 citations

有限長の低密度パリティ検査符号の復号誤り率の解析

Takayuki Nozaki

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.

•Mar 1, 2012•0 citations

Conditional entropy of non-binary LDPC codes over the BEC

Vishwambhar Rathi

We consider transmission over the binary erasure channel (BEC) using non-binary LDPC codes. We generalize the concept of stopping sets to non-binary LDPC codes. We give a combinatorial characterization of decoding failures for non-binary LDPC codes decoded via Belief Propagation (BP). Using the density evolution analysis, we compute the asymptotic residual degree distribution for non-binary LDPC codes. In order to show that asymptotically almost every code in the non-binary LDPC ensemble has a rate equal to the design rate, we generalize the arguments of Measson, Montanari, and Urbanke to the non-binary setting. This generalization enables us to compute the conditional entropy of non-binary LDPC codes. We observe that the Maxwell construction of Measson, Montanari, and Urbanke relating the performance of MAP and BP decoding, holds in the setting of non-binary LDPC codes.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2008•14 citations

Min–Max Decoding for Non-Binary LDPC Codes

Leilei Yang, Fei Liu, Haitao Li

Lecture notes in electrical engineering•Nov 5, 2012•5 citations

Irregular Generalized LDPC codes in Practical Communication Scenarios

Irina E. Bocharova, Boris D. Kudryashov, Sander Mikelsaar

High-rate irregular Generalized LDPC (GLDPC) codes with low-complexity constituent codes are studied. The main area of focus is the optimization of the underlying irregular binary LDPC codes and their matching with constituent codes. Finite length random coding bound on the maximum-likelihood decoding error probability of a random ensemble of irregular GLDPC codes is derived and compared with a similar bound for non-binary (NB) LDPC codes over small alphabets.A new algorithm for assigning columns of the parity-check matrix of a constituent code to nonzero elements of a base matrix of the quasi-cyclic (QC)-GLDPC code is presented. Examples of the constructed QC-GLDPC codes are given. The simulation results for frame error rate (FER) performance of the belief-propagation (BP) decoding for the QC-GLDPC codes are presented and compared with the same performance of NB QC-LDPC codes over small alphabets and binary LDPC codes in the WiFi and 5G standards.Both GLDPC codes and NB LDPC codes with sufficiently low decoding complexity are shown to outperform their binary LDPC counterparts, while GLDPC codes appear to be more advantageous in terms of decoding complexity.

IEEE Information Theory Workshop (ITW)•Nov 1, 2022•1 citation

Message Passing Algorithm with MAP Decoding on Zigzag Cycles for Non-binary LDPC Codes

隆之 野崎, T. Nozaki, 健太 笠井 +3 more

•Jun 1, 2013•0 citations
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.
We consider transmission over the binary erasure channel (BEC) using non-binary LDPC codes. We generalize the concept of stopping sets to non-binary LDPC codes. We give a combinatorial characterization of decoding failures for non-binary LDPC codes decoded via Belief Propagation (BP). Using the density evolution analysis, we compute the asymptotic residual degree distribution for non-binary LDPC codes. In order to show that asymptotically almost every code in the non-binary LDPC ensemble has a rate equal to the design rate, we generalize the arguments of Measson, Montanari, and Urbanke to the non-binary setting. This generalization enables us to compute the conditional entropy of non-binary LDPC codes. We observe that the Maxwell construction of Measson, Montanari, and Urbanke relating the performance of MAP and BP decoding, holds in the setting of non-binary LDPC codes.
High-rate irregular Generalized LDPC (GLDPC) codes with low-complexity constituent codes are studied. The main area of focus is the optimization of the underlying irregular binary LDPC codes and their matching with constituent codes. Finite length random coding bound on the maximum-likelihood decoding error probability of a random ensemble of irregular GLDPC codes is derived and compared with a similar bound for non-binary (NB) LDPC codes over small alphabets.A new algorithm for assigning columns of the parity-check matrix of a constituent code to nonzero elements of a base matrix of the quasi-cyclic (QC)-GLDPC code is presented. Examples of the constructed QC-GLDPC codes are given. The simulation results for frame error rate (FER) performance of the belief-propagation (BP) decoding for the QC-GLDPC codes are presented and compared with the same performance of NB QC-LDPC codes over small alphabets and binary LDPC codes in the WiFi and 5G standards.Both GLDPC codes and NB LDPC codes with sufficiently low decoding complexity are shown to outperform their binary LDPC counterparts, while GLDPC codes appear to be more advantageous in terms of decoding complexity.