Performance Analaysis of Low-Density Parity-Check Codes Derived from Finite Inversive Spaces

Low Density Parity Check (LDPC) codes have been the center of numerous researches in the last ten years. The reason for this interest is due to the performance (Error ratio over SNR) that these codes can achieve keeping the complexity of the pair coding/decoding to a level lower than other codes such as Turbo codes. This project aims to continue and intensify performance simulations on LDPC codes that have been begun in a previous project on the Walton cluster granted by ICHEC. One of its main interests is to study the performance of these codes at bit-error ratios down to 10 , which can only be determined in a long number of Monte-Carlo simulations on a distributed computing cluster with high computational resources. I Context and outcome Reliable communications are in great demand today. Common applications desire higher bandwidth communications in devices consuming less and less energy. It is, therefore, imperative to use the transmission systems available as effectively as possible. The science of finding efficient schemes by which information can be coded for reliable transmission through a noisy channel is called coding theory. The basic idea behind coding and error correction is to add redundant data with each transmission so that, even if errors occur, sufficient protection exists to recover the original message. In other words, the redundant data can be used to recreate information lost during transmission. Many different types of error correcting codes have been discovered during the past 50 years and are used in telecommunications; Among these the BCH and Reed-Solomon (RS) codes are some of the most common codes used from GSM to optical transmissions. In recent years two classes of codes that exhibit performances near the Shannon limit for noisy channels have been developed. These are Turbo codes and Low Density Parity Check codes (LDPC). Introduced by Gallager in 1963 [1] LDPC codes had been neglected until the work of MacKay in 1995[2]; these codes can yield high performances on the binary symmetric channel (BSC) as well as on the additive white Gaussian noise (AWGN) channel, and have been shown to outperform the Turbo codes in many applications. The algorithm used for decoding is called belief propagation, and one of its versions is known as the sum-product algorithm. This algorithm uses a graphical representation of the code, the Tanner Graph. The decoding scheme based on belief propagation is highly efficient, and it is desirable to have the encoding process of these codes as most efficient as well. To achieve this goal various scholars have searched for systematic constructions of LDPC codes. Many good constructions are known nowadays, but only a few of them (if at all) outperform random constructions of LDPC codes. For this reason there is further demand for the systematic construction of LDPC codes with excellent performance. II LDPC Codes from finite Gometry LDPC codes have been systematically constructed in various ways. Margulis [3] initiated the use of a Cayley graph of a group to construct a sparse bipartite graph which in turn induces an LDPC code