Finite Geometry Low Density Parity-Check Codes: Construction, Structure, and Decoding

Abstract

This paper present an algebraic and systematic method for constructing low density parity-check (LDPC) codes from a geometric point of view. LDPC codes are constructed based on the lines and points of a finite geometry. Two classes of LDPC codes are constructed based on the well known Euclidean and projective geometries over finite fields. Codes of these two classes are cyclic and they have good minimum distances and structural properties. They can be decoded in various ways and perform very well with the iterative decoding based on belief propagation (IDBP). These finite geometry LDPC codes can be extended and shortened in various ways. Extension by splitting columns of the parity-check matrices of these codes results in good extended finite geometry LDPC codes. Long extended finite geometry LDPC codes have been constructed and they achieve an error performance only a few tenths of a dB away from the Shannon limit with IDBP. Finite geometry LDPC codes are strong competitors to turbo codes for error control in communication and digital data storage systems.