New Classes of Partial Geometries and Their Associated LDPC Codes

TL;DR

This paper introduces a new simple construction of quasi-cyclic partial geometries for LDPC codes, investigates their trapping sets, and explores their potential as good expanders for improved decoding performance.

Contribution

It presents a novel, straightforward method to construct quasi-cyclic partial geometries and analyzes their properties, including trapping sets and expansion characteristics.

Findings

New quasi-cyclic partial geometries constructed

Analysis of trapping sets in these geometries

Potential for good expansion properties in Tanner graphs

Abstract

The use of partial geometries to construct parity-check matrices for LDPC codes has resulted in the design of successful codes with a probability of error close to the Shannon capacity at bit error rates down to $10^{-15}$. Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with quasi-cyclic structure is given and their properties are investigated. The trapping sets of the partial geometry codes were considered previously 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 under message-passing decoding.