The 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.

有限射影空间子集的线性表示是由子集确定的仿射点和线的入射系统。在本文中，我们使用特征理论来证明入射矩阵的秩对某些超平面具有直接的几何解释。我们认为以入射矩阵及其转置为奇偶校验矩阵定义的LDPC码，在前一种情况下证明了范登德里斯猜想，即该码是由权重最小的单词（称为平面单词）生成的。在后一种情况下，我们在几种情况下计算最小权重，并提供最小权重码字的显式构造。