LDPC codes associated with linear representations of geometries

Abstract

We look at low density parity check codes over a finite field $\mathbb K$ associated with finite geometries $T$2*$(\mathcal K)$, where $\mathcal K$ is any subset of PG$(2,q)$, with $q=p$h, $p$≠char$\mathbb K$. This includes the geometry $LU(3,q)$D, the generalized quadrangle $T$2*$(\mathcal K)$ with $\mathcal K$ a hyperoval, the affine space AG$(3,q)$ and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of $\mathcal K$.