Tomoharu Shibuya

Weight and Stopping Set Distributions of Two-Edge Type LDPC Code Ensembles

In this paper, we explicitly formulate the average stopping set distributions and their asymptotic exponents of two instances of two-edge type LDPC code ensembles. Further we investigate the relation between the asymptotic exponents of those two code ensembles

Detailed representation of irregular LDPC code ensembles and density evolution

We construct new families of low-density parity-check (LDPC) code ensembles which are subensembles of conventional irregular LDPC code ensembles and develop density evolution for those ensembles. Further, we find an ensemble which has a better threshold than that of its super ensemble.

Weight distribution of non-binary LDPC codes

Weight distributions of binary low-density parity-check (LDPC) codes are well studied. We investigate the average distributions of symbol and binary weight for non-binary LDPC code ensemble. We derive the asymptotic growth rate and its linear approximation for small weight. Moreover, we show the typical minimum distance does not monotonically increase with the size of the field where the codes are defined.

Average Coset Weight Distribution of Multi-Edge Type LDPC Code Ensembles

Multi-Edge type Low-Density Parity-Check codes (MET-LDPC codes) introduced by Richardson and Urbanke are generalized LDPC codes which can be seen as LDPC codes obtained by concatenating several standard (ir)regular LDPC codes. We prove in this paper that MET-LDPC code ensembles possess a certain symmetry with respect to their Average Coset Weight Distributions (ACWD). Using this symmetry, we drive ACWD of MET-LDPC code ensembles from ACWD of their constituent ensembles.

Two-edge type LDPC code ensembles with exponentially few codewords with linear small weight

Multiedge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles has better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. We derive the condition corresponding to Orlitskypsilas condition for two-edge type LDPC code ensembles which is simple example of Multi-Edge type LDPC code ensembles.

Performance of Standard Irregular LDPC Codes under Maximum Likelihood Decoding

In this paper, we derive an upper bound for the average block error probability of a standard irregular low-density parity-check (LDPC) code ensemble under the maximum-likelihood (ML) decoding. Moreover, we show that the upper bound asymptotically decreases polynomially with the code length. Furthermore, when we consider several regular LDPC code ensembles as special cases of standard irregular ones over an additive white Gaussian noise channel, we numerically show that the signal-to-noise ratio (SNR) thresholds at which the proposed bound converges to zero as the code length tends to infinity are smaller than those for a bound provided by Miller et al. We also give an example of a standard irregular LDPC code ensemble which has a lower SNR threshold than a given regular LDPC code ensemble.

Asymptotic bit error probability of LDPC codes for the binary erasure channel with finite number of iterations

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. Furthermore, a gap between the bit error probability after finite number of iterations for finite block length n and that for infinite block length is asymptotically alpha/n, where alpha denotes a speci..c constant determined by a degree distribution, a number of iterations and erasure probability. Our main result is to derive an ef..cient algorithm for calculating alpha for regular ensembles.

Block-triangularization of parity check matrices for efficient encoding of linear codes

In this paper, we propose a new encoding algorithm for linear codes whose computational complexity is O(w(H)) where w(H) denotes the number of non-zero elements in a parity check matrix H of a code. The proposed algorithm is based on the block-triangularization — an efficient technique to solve a system of linear equations — of a parity part of a parity check matrix, combining additional row and column permutations. As a result, the proposed algorithm can encode any linear codes defined by sparse parity check matrices, such as LDPC codes, with O(n) complexity where n denotes the code length.

Analysis of weight distributions of two-edge type LDPC codes by Hayman approximation

Multi-edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles have better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with small linear weight codewords exponentially decreasing in code length. Nakasendo et al. derived the condition in which code ensembles have exponentially decreasing small linear weight codewords for two-edge type LDPC code ensembles which is simple example of multi-edge type LDPC code ensembles. Our conclusion is the same as that of Nakasendo et al., and although Nakasendopsilas method is difficult to apply to more than three edge-types, there is a possibility that our method using Hayman approximation as Di et al. do derives the condition for multi-edge type LDPC codes ensembles.

Asymptotic gaps between BP decoding and local-MAP decoding for low-density parity-check codes

In this paper, we consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes. We consider MAP decoding not using a whole Tanner graph but only a neighborhood graph of fixed depth referred to as local-MAP decoding for deriving lower bounds of the error probability under message-passing decoding and bit-flipping decoding. The main result of this paper is to derive an asymptotic performance for regular ensembles under local-MAP decoding and to derive an asymptotic gap of the bit error probability between belief propagation (BP) and local-MAP decoding for irregular ensembles. Finally, we show the limit of the scaling parameter of these decodings when number of iterations tends to infinity.