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arXiv:cs/0504016 · IEEE Transactions on Information Theory

Shortened Array Codes of Large Girth

Olgica Milenković, Navin Kashyap, David Leyba

Published Jul 26, 2006• 113 citations
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Abstract

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio (SNR) in the case of communication over an additive white Gaussian noise (AWGN) channel.

DOI
10.1109/tit.2006.878179

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One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio (SNR) in the case of communication over an additive white Gaussian noise (AWGN) channel.
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.
One approach for designing the one-coincidence sequence (OCS) low-density parity-check (LDPC) codes of large girth is investigated. These OCS-LDPC codes are quasi-cyclic, and their parity-check matrices are composed of circulant permutation matrices. Generally, the cycle structures in these codes are determined by the shift values of circulant permutation matrices, and the existence of cycles in the corresponding Tanner graph is governed by certain cycle-governing equations (CGEs). Therefore, finding the proper shift values is the key point to increase the girth of these codes. In this paper, we provide an effective method to systematically find out the CGEs for these codes of girth 6, 8, and 10, respectively. Then, one less computation-intensive algorithm is used to generate the proper shift values for constructing the OCS-LDPC codes of large girth. Simulation results show that significant gains in signal-to-noise ratio over an additive white-Gaussian noise channel can be achieved by increasing the girth of the OCS-LDPC codes.
Low Density Parity Check (LDPC) codes are specified by the parity check matrix H. Smaller girths in the Tanner graphs of LDPC codes prevent the sum-product algorithm from converging, i.e., short cycles in H-matrix degrade the performance of LDPC decoder. In this paper, we present an algorithm for constructing LDPC codes without short cycles. The H-matrix must be sufficiently sparse to increase the girth of a code. In this work, we present a simple method to increase the sparsity of matrix and provide lemmas to avoid short cycles of lengths 4 and 6. We design sub-matrices for column-weight two and column-weight three codes. This design makes use of a simple shifting function given by the proposed scheme. These are then combined into a final H-matrix using Gray code representations. The proposed method provides structured, regular codes with flexibility in code rates and code lengths. The regularity of these codes gives the advantages like simplicity of hardware implementation and fast encoding. Key words LDPC codes, Tanner Graph, Girth, Gray codes
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
This paper gives necessary and sufficient conditions for the Tanner graph of a quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-one protograph to have girth 6, 8, 10, and 12, respectively, in the case of parity-check matrices with column weight 4. These results are a natural extension of the girth results of the already-studied cases of column weight 2 and 3, and it is based on the connection between the girth of a Tanner graph given by a parity-check matrix and the properties of powers of the product between the matrix and its transpose. The girth conditions can be easily incorporated into fast algorithms that construct codes of desired girth between 6 and 12; our own algorithms are presented for each girth, together with constructions obtained from them and corresponding computer simulations. More importantly, this paper emphasizes how the girth conditions of the Tanner graph corresponding to a parity-check matrix composed of circulants relate to the matrix obtained by adding (over the integers) the circulant columns of the parity-check matrix. In particular, we show that imposing girth conditions on a parity-check matrix is equivalent to imposing conditions on a square circulant submatrix obtained from it.