Low Density Parity Check Codes

Abstract

The Tanner graph is described. A girth in the Tanner graph is equivalent to a short cycle of 1- symbols in the parity check matrix. The condition called row-column constraint is introduced, in order to allow practical decoding procedures. They are based on the sum-product algorithm, which is briefly outlined. Regular and irregular LDPC codes are introduced. The firstly proposed LDPC codes by Gallager and MacKay-Neal are reviewed. The following main families of more recent LDPC codes are briefly described: codes based on protographs, codes constructed employing repeat and accumulation devices, codes derived from the decomposition of finite geometries, codes obtained starting from MDS codes. Furthermore codes obtained from superimposition or by circulant expansion are analysed. Masking and row or column splitting can be employed for reducing the 1-symbol density and breaking short cycles. The effects of such procedures on the code rate are stressed. For instance, circulant expansion and masking do not vary the code rate, whereas row (column) splitting increases (reduces) it. Irregular LDPC codes can be designed by means of proper rules progressively adding edges to the Tanner graph. A statistical treatment, called density evolution, is presented, in order to obtain asymptotic best performance at different code rates, taking into account the intrinsic nature of the sum-product decoding algorithm. A first approach to LDPC convolutional codes is presented, starting from good known LDPC block codes. The use of unwrapping is suggested, in particular with suitable QC codes, or a derivation from properly modified H-extension is reviewed.