# Linear representations of finite geometries and associated LDPC codes

## TL;DR

This paper explores the linear representations of finite geometries, analyzing their incidence matrices, and studies associated LDPC codes, proving conjectures and computing minimum weights for specific cases.

## Contribution

It introduces a geometric interpretation of the incidence matrix rank and proves a conjecture about the generation of LDPC codes by plane words.

## Key findings

- The rank of the incidence matrix relates to hyperplanes in the geometry.
- LDPC codes are generated by minimum weight words called plane words.
- Minimum weight codewords are explicitly constructed in several cases.

## Abstract

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.