  The {\it linear representation} of a subset of a finite projective space is
an incidence system of affine points and lines determined by the subset. In
this paper we use character theory to show that the rank of the incidence
matrix has a direct geometric interpretation in terms of certain hyperplanes.
We consider the LDPC codes defined by taking the incidence matrix and its
transpose as parity-check matrices, and in the former case prove a conjecture
of Vandendriessche that the code is generated by words of minimum weight called
plane words. In the latter case we compute the minimum weight in several cases
and provide explicit constructions of minimum weight codewords.
