Low-density parity-check codes based on finite geometries: A rediscovery and new results

This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed base d on the lines and points of Euclidean and projective geometries over finit e fields. Codes of these four classes have good minimum distances and their Tanner graphs have girth 6. Finite-geometry LDPC codes can be decoded in v arious ways, ranging from low to high decoding complexity and from reasonab ly good to very good performance. They perform very well with iterative dec oding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by othe r LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LD PC codes. Several techniques of extension and shortening are presented. Lon g extended finite-geometry LDPC codes have been constructed and they achiev e a performance only a few tenths of a decibel away from the Shannon theore tical limit with iterative decoding.