LDPC Codes Which Can Correct Three Errors Under Iterative Decoding

In this paper, we provide necessary and sufficient conditions for a column-weight-three LDPC code to correct three errors when decoded using Gallager A algorithm. We then provide a construction technique which results in a code satisfying the above conditions. We also provide numerical assessment of code performance via simulation results.Comment: 5 pages, 3 figures, submitted to IEEE Information Theory Workshop (ITW), 200

Multi-Stream LDPC Decoder on GPU of Mobile Devices

Low-density parity check (LDPC) codes have been extensively applied in mobile communication systems due to their excellent error correcting capabilities. However, their broad adoption has been hindered by the high complexity of the LDPC decoder. Although to date, dedicated hardware has been used to implement low latency LDPC decoders, recent advancements in the architecture of mobile processors have made it possible to develop software solutions. In this paper, we propose a multi-stream LDPC decoder designed for a mobile device. The proposed decoder uses graphics processing unit (GPU) of a mobile device to achieve efficient real-time decoding. The proposed solution is implemented on an NVIDIA Tegra board as a system on a chip (SoC), where our results indicate that we can control the load on the central processing units through the multi-stream structure

On the guaranteed error correction capability of LDPC codes

We investigate the relation between the girth and the guaranteed error correction capability of $\gamma$-left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms. A lower bound on the number of variable nodes which expand by a factor of at least $3 \gamma/4$ is found based on the Moore bound. An upper bound on the guaranteed correction capability is established by studying the sizes of smallest possible trapping sets.Comment: 5 pages, submitted to IEEE International Symposium on Information Theory (ISIT), 200

Error Correction Capability of Column-Weight-Three LDPC Codes: Part II

The relation between the girth and the error correction capability of column-weight-three LDPC codes is investigated. Specifically, it is shown that the Gallager A algorithm can correct $g/2-1$ errors in $g/2$ iterations on a Tanner graph of girth $g \geq 10$.Comment: 7 pages, 7 figures, submitted to IEEE Transactions on Information Theory (July 2008

On Trapping Sets and Guaranteed Error Correction Capability of LDPC Codes and GLDPC Codes

The relation between the girth and the guaranteed error correction capability of $\gamma$-left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms is investigated. A lower bound on the size of variable node sets which expand by a factor of at least $3 \gamma/4$ is found based on the Moore bound. An upper bound on the guaranteed error correction capability is established by studying the sizes of smallest possible trapping sets. The results are extended to generalized LDPC codes. It is shown that generalized LDPC codes can correct a linear fraction of errors under the parallel bit flipping algorithm when the underlying Tanner graph is a good expander. It is also shown that the bound cannot be improved when $\gamma$ is even by studying a class of trapping sets. A lower bound on the size of variable node sets which have the required expansion is established.Comment: 17 pages. Submitted to IEEE Transactions on Information Theory. Parts of this work have been accepted for presentation at the International Symposium on Information Theory (ISIT'08) and the International Telemetering Conference (ITC'08

Desain Arsitektur dan Implementasi Pengkode-Pendekode Hard Decision LDPC Menggunakan Algoritma Message Passing

Low Density Parity Check atau LDPC merupakan teknik channel coding yang dikembangkan oleh Robert G. Gallager pada tahun 1962. LDPC dikenal akan kemampuannya sebagai teknik pengkode yang terdekat untuk mencapai maksimum dari Shannon capacity. Oleh karena itu, LDPC sangat tepat untuk digunakan pada high data rate dan high channel noise. Selain kemampuan untuk mencapai kapasitas maksimum Shannon, LDPC••••••••••• juga dikenal sebagai teknik channel coding yang memiliki kompleksitas pada implementasi yang rendah. Tugas akhir ini bertujuan untuk merancang arsitektur dan mengimplementasikan teknik pengkode dan pendekode LDPC pada FPGA Cyclone II. Metode koreksi error yang digunakan pada implementasi adalah algoritma message passing. Proses implementasi menggunakan dua jenis code rate yaitu code rate ½ dan code rate ¾. Pada code rate ½, matriks yang digunakan adalah matriks 4x8, matriks 8x16, dan matriks 24x48. Pada code rate ¾, matriks yang digunakan adalah matriks 4x16. Hasil pengujian menunjukkan sistem berhasil diimplementasikan dengan baik pada FPGA Cyclone II. Implementasi sistem dengan algoritma message passing dapat melakukan koreksi error untuk lebih dari satu bit error. Frekuensi kerja matriks 4x8 adalah 1,35 MHz. Frekuensi kerja matriks 8x16 adalah 0,909 MHz. Frekuensi kerja matriks 24x48 adalah 0,156 MHz. Frekuensi kerja matriks 4x16 adalah 1,35 MHz. Kata kunci : Code rate, FPGA Cyclone II, LDPC, message passin

Estudio de códigos finitos LDPC y desarrollo de una herramienta simple de diseño

Los códigos LDPC (Low-Density Parity-Check) son los códigos correctores cuyas prestaciones son las que más se acercan a los límites teóricos que estableció Shannon. Debido a esta propiedad, se emplean en una gran variedad de sistemas de comunicaciones. La característica más importante de estos códigos radica en que sus matrices de chequeo de paridad son de baja densidad, es decir, tienen pocos elementos distintos de 0 y por esta razón, tanto la codificación como la decodificación mediante ellos es muy eficiente en términos de complejidad computacional. Existen diversos métodos de decodificación para estos códigos, pero el más utilizado es el algoritmo belief propagation, también conocido como suma-producto, que se basa en el intercambio de información probabilística estimada para cada bit en cada fila de chequeo de paridad de H. Con este estudio se quiere comprobar que el rendimiento mostrado por códigos LDPC cuando la longitud de los códigos tiende a infinito tiene un comportamiento que puede extrapolarse al caso en que los códigos tienen una longitud finita. Para ello estudiaremos estos códigos en un escenario muy sencillo donde podremos extraer un modelo de cómo se comporta el decodificador para códigos finitos.The LDPC codes (Low-Density Parity-Check) are correcting codes whose benefits are those that come closest to the theoretical limits established Shannon. Because of this property, they are used in a variety of communications systems. The most important characteristic of these codes is that the parity check matrixes are low density, that is, they have few elements different from 0, and for this reason, both encoding and decoding using them is very efficient in terms of computational complexity. There are several methods of decoding these codes, but the most used is the belief propagation algorithm, also known as the sum-product, which is based on the exchange of probabilistic information estimated for each bit in each parity check row from H. With this study we want to verify that the performance shown by LDPC codes when the code length tends to infinity has a behavior that can be extrapolated to the case where the codes have a finite length. We will study these codes in a simple scenario where we can extract a model of how the decoder behaves for finite codes.Ingeniería de Sistemas de Comunicacione

Optimal finite alphabet sources over partial response channels

We present a serially concatenated coding scheme for partial response channels. The encoder consists of an outer irregular LDPC code and an inner matched spectrum trellis code. These codes are shown to offer considerable improvement over the i.i.d. capacity (> 1 dB) of the channel for low rates (approximately 0.1 bits per channel use). We also present a qualitative argument on the optimality of these codes for low rates. We also formulate a performance index for such codes to predict their performance for low rates. The results have been verified via simulations for the (1-D)/sqrt(2) and the (1-D+0.8D^2)/sqrt(2.64) channels. The structure of the encoding/decoding scheme is considerably simpler than the existing scheme to maximize the information rate of encoders over partial response channels