New Classes of Partial Geometries and Their Associated LDPC Codes

The use of partial geometries to construct parity-check matrices for binary low-density parity-check (LDPC) codes has resulted in the design of successful codes with a probability of error on the AWGN channel close to the Shannon capacity at bit error rate down to $10^{-15}$ . Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with a quasi-cyclic (QC) structure is given and their properties are investigated. Two new classes of this type of partial geometries, one based on prime fields and the other based on cyclic subgroups of prime orders of finite fields, are constructed. QC-LDPC codes with good error performances are constructed based on these two new classes of partial geometries. The trapping sets of the partial geometry codes were previously considered using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets. This topic is further considered here. Finally, there is a natural relationship between partial geometries and strongly regular graphs. The eigenvalues of the adjacency matrices of such graphs are well known, and it is of interest to determine if any of the Tanner graphs derived from the partial geometries are good expanders for certain parameter sets, since it can be argued that codes with good geometric and expansion properties might perform well on the AWGN channel under message-passing decoding.