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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Spatially “Mt. Fuji” Coupled LDPC Codes

Yuta Nakahara, Shota Saito, Toshiyasu Matsushima

Published Jan 1, 2017• 3 citations
Publisher

Abstract

A new type of spatially coupled low density parity check (SCLDPC) code is proposed. This code has two benefits. (1) This code requires less number of iterations to correct the erasures occurring through the binary erasure channel in the waterfall region than that of the usual SCLDPC code. (2) This code has lower error floor than that of the usual SCLDPC code. Proposed code is constructed as a coupled chain of the underlying LDPC codes whose code lengths exponentially increase as the position where the codes exist is close to the middle of the chain. We call our code spatially “Mt. Fuji” coupled LDPC (SFCLDPC) code because the shape of the graph representing the code lengths of underlying LDPC codes at each position looks like Mt. Fuji. By this structure, when the proposed SFCLDPC code and the original SCLDPC code are constructed with the same code rate and the same code length, L (the number of the underlying LDPC codes) of the proposed SFCLDPC code becomes smaller and M (the code lengths of the underlying LDPC codes) of the proposed SFCLDPC code becomes larger than those of the SCLDPC code. These properties of L and M enables the above reduction of the number of iterations and the bit error rate in the error floor region, which are confirmed by the density evolution and computer simulations.

DOI
10.1587/transfun.e100.a.2594

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A new type of spatially coupled low density parity check (SCLDPC) code is proposed. This code has two benefits. (1) This code requires less number of iterations to correct the erasures occurring through the binary erasure channel in the waterfall region than that of the usual SCLDPC code. (2) This code has lower error floor than that of the usual SCLDPC code. Proposed code is constructed as a coupled chain of the underlying LDPC codes whose code lengths exponentially increase as the position where the codes exist is close to the middle of the chain. We call our code spatially “Mt. Fuji” coupled LDPC (SFCLDPC) code because the shape of the graph representing the code lengths of underlying LDPC codes at each position looks like Mt. Fuji. By this structure, when the proposed SFCLDPC code and the original SCLDPC code are constructed with the same code rate and the same code length, L (the number of the underlying LDPC codes) of the proposed SFCLDPC code becomes smaller and M (the code lengths of the underlying LDPC codes) of the proposed SFCLDPC code becomes larger than those of the SCLDPC code. These properties of L and M enables the above reduction of the number of iterations and the bit error rate in the error floor region, which are confirmed by the density evolution and computer simulations.
Spatially “Mt. Fuji” coupled (SFC) low density parity check (LDPC) codes are constructed as a chain of block LDPC codes. A difference between the SFC-LDPC codes and the original spatially coupled (SC) LDPC codes is code lengths of the underlying block LDPC codes. The code length of the block LDPC code at the middle of the chain is larger than that at the end of the chain. It is experimentally confirmed that the bit error probability in the error floor region of the SFC-LDPC code is lower than that of the SC-LDPC code whose code length and design rate are the same as those of the SFC-LDPC code. In this letter, we calculate the weight distribution of the SFC-LDPC code and try to explain causes of the low bit error rates of the SFC-LDPC code.
A spatially “Mt. Fuji” coupled low-density parity check (LDPC) ensemble is a modified version of the original spatially coupled (SC) LDPC ensemble. Its desirable properties are first observed in experimentally. The decoding error probability in the error floor region over the binary erasure channel (BEC) is theoretically analyzed later. In this paper, as the last piece of the theoretical analysis over the BEC, we analyze the decoding error probability in the waterfall region by modifying the covariance evolution which has been used to analyze the original SC-LDPC ensemble.
A spatially “Mt. Fuji” coupled (SFC) low-density parity-check (LDPC) ensemble is a modified version of the spatially coupled (SC) LDPC ensemble. Its decoding error probability in the waterfall region has been studied only in an experimental manner. In this paper, we theoretically analyze it over the binary erasure channel by modifying the expected graph evolution (EGE) and covariance evolution (CE) that have been used to analyze the original SC-LDPC ensemble. In particular, we derive the initial condition modified for the SFC-LDPC ensemble. Then, unlike the SC-LDPC ensemble, the SFC-LDPC ensemble has a local minimum on the solution of the EGE and CE. Considering the property of it, we theoretically expect the waterfall curve of the SFC-LDPC ensemble is steeper than that of the SC-LDPC ensemble. In addition, we also confirm it by numerical experiments.
The finite-length performance of multi-edge spatially coupled low-density parity-check (SC-LDPC) codes over the binary erasure channel (BEC) is analyzed. Existing scaling laws are extended to arbitrary protograph base matrices that include puncturing patterns and multiple edges between nodes. A regular protograph-based SC-LDPC construction based on the (4; 8)-regular LDPC block code works well in the waterfall region compared to more involved rate-1/2 structures proposed to improve the threshold to minimum distance trade-off. Scaling laws are also used for code design and to estimate the block length of a given SC-LDPC code ensemble to match the performance of some other code. Estimates on the performance degradation are developed if the chain length varies.
The finite length performance of codes on graphs constructed by connecting spatially coupled low-density parity-check (SC-LDPC) code chains is analyzed. Successive (peeling) decoding is considered for the binary erasure channel (BEC). The evolution of the undecoded portion of the bipartite graph remaining after each iteration is analyzed as a dynamical system. It is shown that, in addition to superior iterative decoding thresholds, connected chain ensembles have better performance than single chain ensembles of the same rate and length.
For designing low-density parity-check (LDPC) codes for quantum error-correction, we desire to satisfy the conflicting requirements below simultaneously. 1) The row weights of parity-check “should be large”: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices “should not be large”: The performance of the sum-product decoding algorithm at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled (SC) LDPC codes exhibit capacity-achieving performance for classical channels. SC LDPC codes have both large row weight and capacity-achieving error-floor and water-fall performance. In this paper, we propose a new class of quantum LDPC codes based on spatially coupled quasi-cyclic LDPC codes. The performance outperforms that of quantum “non-coupled” quasi-cyclic LDPC codes.