We detail for the first time a complete explicit description of the quasi-cyclic structure of all classical finite generalized quadrangles. Using these descriptions we construct families of quasi-cyclic LDPC codes derived from the point-line incidence matrix of the quadrangles by explicitly calculating quasi-cyclic generator and parity check matrices for these codes. This allows us to construct parity check and generator matrices of all such codes of length up to 400000. These codes cover a wide range of transmission rates, are easy and fast to implement and perform close to Shannon's limit with no visible error floors. We also include some performance data for these codes. Furthermore, we include a complete explicit description of the quasi-cyclic structure of the point-line and point-hyperplane incidences of the finite projective and affine spaces.

翻译：我们首次完整且显式地描述了所有经典有限广义四边形的准循环结构。利用这些描述，我们通过显式计算这些码的准循环生成矩阵和校验矩阵，构建了源自四边形点线关联矩阵的准循环LDPC码族。这使得我们能够构建所有此类码长不超过400000的校验矩阵和生成矩阵。这些码覆盖了广泛的传输速率范围，易于快速实现，性能接近香农极限且无明显错误平层。我们还提供了这些码的部分性能数据。此外，我们完整显式地描述了有限射影空间与仿射空间中点线关联及点超平面关联的准循环结构。