Low density parity check codes

Abstract

Low Density Parity Check codes, LDPCs for short, are a family of codes which have shown near optimal error-correcting capabilites. They were proposed in 1963 by Robert Gallager in his PhD thesis. While he proved that probabilistic constructions of random LDPCs gave asymptotically good linear codes, they were largely abandoned due to the lack of computing power to make them practically feasable. They enjoyed a re-birth during the coding revolution of the 1980's, and thanks to the developement of expander graph theory, it was proven that they can be encoded and decoded in linear time. This thesis will review the main results through this journey. Nowadays, LDPCs appear in a plethora of commercial applications. The codes used in practice and the techniques that were employed to construct them will also be explored in this work. Finally, a new family of LDPCs will be proposed, which will be constructed from incidence structures called generalized quadrangles, and perform markedly better than random codes.