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International Conference on Communications

Geometry based designs of LDPC codes

Haotian Zhang, José M. F. Moura

Published Jan 1, 2004• 4 citations
Publisher

Abstract

In this paper we construct three types of low-density parity-check codes with column weight j = 3 based on geometries in graphical models. Low-density parity-check codes with j > 2 are desired because their minimum distance improves linearly with the code block length n. The codes we present here have girth 8 and girth 10. All codes are regular and well-structured. These codes have flexible block lengths and code rates, and may be used in the area of communications and data storage. Our simulation results show that they have better bit-error-rate decoding performance and lower error floors in additive white Gaussian noise channels than randomly constructed low-density parity-check codes.

DOI
10.1109/icc.2004.1312604

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In this paper we construct three types of low-density parity-check codes with column weight j = 3 based on geometries in graphical models. Low-density parity-check codes with j > 2 are desired because their minimum distance improves linearly with the code block length n. The codes we present here have girth 8 and girth 10. All codes are regular and well-structured. These codes have flexible block lengths and code rates, and may be used in the area of communications and data storage. Our simulation results show that they have better bit-error-rate decoding performance and lower error floors in additive white Gaussian noise channels than randomly constructed low-density parity-check codes.
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In this paper, some well-structured block designs are presented whose incidence matrices can be considered as the mother matrices of some regular and irregular column-weight three quasi cyclic low-density parity-check codes of girth 20. The class of LDPC codes derived from our construction possesses much more flexibility in terms of code length and rate. Moreover, the length of the regular constructed codes with rate at most 0.5 is at least 4.9 times smaller than the length of the voltage graph LDPC codes with the same rate and girth. Simulation results show that the constructed codes outperform progressive edge growth, repeat accumulate and random LDPC codes.
Four classes of maximum-girth geometrically structured column-weight-two regular quasi-cyclic (QC) low-density parity-check (LDPC) codes are introduced. Two classes of these codes, referred to as Type-I and Type-II codes, are with row-weights 4 and 3, and maximum girths 16 and 24, respectively. The idea behind the construction of these two classes of codes, with rates at least 1/2 and 1/3, is slightly generalized to obtain two classes of variable-high-rate codes, referred to as Type-III1 and Type-III2 codes, with maximum girth 20 and 16, respectively. A low-complexity deterministic algorithm for constructing these four classes of codes is given. The algorithm generates maximum-girth Type-I and Type-II codes with almost arbitrary length eta not less than 216 and 243, respectively. The output of the algorithm substantially improves on some of the previously best known codes constructed using a randomized progressive edge-growth (RPEG) algorithm. For instance, we have rate-0.71 Type-III1 codes of lengths 308 and 728 with girths 10 and 12, respectively, versus the code lengths 385 and 840 obtained by the RPEG algorithm. Simulation results on AWGN channel confirm that, from BER performance perspective, the constructed LDPC codes are superior to the column-weight-two LDPC codes constructed by the previously reported methods. The generator matrix G(D) of the convolutional codes associated with Type-I and Type-II codes is given. The free distance dfreeof such a convolutional code is equal to the minimum distance of the corresponding QC block code.
This article describes the different methods to design regular low density parity-check (LDPC) codes with large girth. In graph terms, this corresponds to designing bipartite undirected regular graphs with large girth. Large girth speeds the convergence of iterative decoding and improves the performance at least in the high SNR range, by slowing down the onsetting of the error floor. We reviewed several existing constructions from exhaustive search to highly structured designs based on Euclidean and projective finite geometries and combinatorial designs. We describe GB and TS LDPC codes and compared the BER performance with large girth to the BER performance of random codes. These studies confirm that in the high SNR regime these codes with high girth exhibit better BER performance. The regularity of the codes provides additional advantages that we did not explore in this article like the simplicity of their hardware implementation and fast encoding.
We present a novel construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with arbitrary column weight and girth at least eight. By carefully designing the parity-check matrix, our approach ensures both structural flexibility and improved error-correction capabilities. We evaluate the performance of these codes under the additive white Gaussian noise (AWGN) channel. Simulation results demonstrate that the proposed QC-LDPC codes outperform the protograph LDPC codes included in the Consultative Committee for Space Data Systems (CCSDS) telecommand standards. Additionally, we benchmark our proposed codes against existing short-length codes with girth 8, observing improvements in the decoding performance while ensuring suitability for hardware implementation. The findings of this work contribute to the ongoing development of efficient LDPC codes suitable for ultra-reliable low-latency communications and area-efficient architecture designs for the Internet of Things (IoT), machine-to-machine (M2M), and satellite communications.