The pseudo-hyperplanes and homogeneous pseudo-embeddings of the generalized quadrangles of order (3, t)

In the paper [8], we introduced the notions of pseudo-hyperplane and pseudoembedding of a point-line geometry and proved that every generalized quadrangle of order (s, t), 2 ≤ s <∞, has faithful pseudo-embeddings. The present paper focuses on generalized quadrangles of order (3, t). Using the computer algebra system GAP [12] and invoking some theoretical relationships between pseudo-hyperplanes and pseudo-embeddings obtained in [8], we are able to give a complete classification of all pseudo-hyperplanes of Q. We hereby find several new examples of tight sets of generalized quadrangles, as well as a complete classification of all 2-ovoids of Q. We use the classification of the pseudo-hyperplanes of Q to obtain a list of all homogeneous pseudo-embeddings of Q.