Strongly regular generalized partial geometries and associated LDPC codes
Lijun Ma, Changli Ma, Zihong Tian
TL;DR
This work introduces strongly regular generalized partial geometries (SRPGs) of grade $r$, unifying partial geometries and $(\alpha,\beta)$-geometries, and develops two grade-3 constructions from conics in $PG(2,q)$ and hyperbolic quadrics in $PG(3,q)$. It defines LDPC codes from SRPG incidence matrices, derives minimum-distance bounds via spectral analysis, and analyzes $2$-ranks and girth properties, with empirical performance suggesting improved error-correction over random LDPC codes. The results advance structured LDPC design through geometric incidence structures, providing explicit parameter sets, conjectures on $2$-rank behavior, and practical decoding performance insights. The combination of exact SRPG constructions and LDPC performance evaluation highlights the potential of geometric combinatorics for robust coding solutions.
Abstract
In this paper, we introduce strongly regular generalized partial geometries of grade $r$, which generalise partial geometries and strongly regular $(α,β)$-geometries. By the properties of quadrics in PG$(2,q)$ and PG$(3,q)$, we construct two classes of strongly regular generalized partial geometries of grade $3$. Besides, we define low-density parity-check (LDPC) codes by considering the combinatorial structures of strongly regular generalized partial geometries and derive bounds on minimum distance, dimension and girth for the LDPC codes.