LDPC codes from triangle-free line sets
Let \(AG(n,q)\) and \(PG(n,q)\) be the finite affine and projective spaces respectively of dimension \(n\) over \(GF(q)\). A subset \(\mathcal L\) of lines of \(AG(n,q)\) or \(PG(n,q)\) is said to be triangle-free if no three nonconcurrent lines of \(\mathcal L\) form a triangle. If \(G\) is the bipartite graph whose vertices are the points of a finite space and whose edges are determined by natural inclusion, then the point-line graph associated with a triangle-free set turns out to have no cycles of length \(6\) and girth at least \(8\). It follows that the associated incidence matrix is relatively sparse and so can be used to construct a so-called low-density parity check code. In the paper under review the author gives two constructions of triangle-free sets and hence the parameters of the associated codes when \(q\) is sufficiently small.
scientific article; zbMATH DE number 2091526