The homogeneous pseudo-embeddings and hyperovals of the generalized quadrangle \(H(3,4)\) (Q2310391)

Let \(\mathcal{S}=(\mathcal{P},\mathcal{L})\) be a generalized quadrangle and denote by \(\varepsilon:\mathcal{S} \longrightarrow \mathrm{PG}(V)\) the universal pseudo-embedding \(\mathcal{S}.\) The vector dimension of the subspace \(\langle \varepsilon(\mathcal{P})\rangle\) of \(V\) is called the \textit{pseudo-embedding rank} of \(\mathcal{S}.\) If \(|\mathcal{P}|\) is finite, denote by \(C=C(\mathcal{S})\) the binary code of length \(|\mathcal{P}|\) generated by the characteristic vectors of the lines of \(\mathcal{S}.\) It has been proved in [the first author, Adv. Geom. 13, No. 1, 71--95 (2013; Zbl 1267.51002)] that the pseudo-embedding rank of \(\mathcal{S}\) is \(|\mathcal{P}|- \dim(C).\) In this paper, the authors prove that the pseudo-embedding rank of the Hermitian quadrangle \(H(3,4)\) is equal to \(24.\) As a consequence, the binary code \(C(H(3,4))\) has dimension \(45-24=21,\) because the generalized quadrangle \(H(3,4)\) has \(45\) points. They also show that there are, up to isomorphism, four homogeneous pseudo-embeddings of \(H(3,4),\) with respective vector dimensions \(14,\) \(15,\) \(23\) and \(24.\)