Cyclic Partial Geometries and Their Associated LDPC and Constant-Weight Codes

Partial geometries form an interesting branch in combinatorial mathematics, and recently they have been shown to be very effective in the construction of LDPC codes with distinct geometric and algebraic structures. This paper presents three specific cyclic classes of partial geometries. Based on these three classes of partial geometries, three classes of LDPC codes and three classes of constant-weight codes are constructed. Codes in these two categories are either cyclic or quasi-cyclic. Designs and constructions of these codes are straightforward and flexible without a need for extensive computer search. It is shown that long high-rate LDPC codes constructed based on the three classes of cyclic partial geometries perform well over the additive white Gaussian noise channel (AWGNC) with iterative decoding algorithm based on belief propagation. They can achieve low error-rates without visible error-floor and their decoding converges rapidly. These LDPC codes also perform well over the binary erasure channel (BEC) and are very effective in correcting phased-bursts of erasures. Based on cyclic partial geometries, a special type of constant-weight codes, called balanced codes, can be constructed. The constant-weight codes constructed based on partial geometries are either optimal or nearly optimal.