Extended \(4 \times{} 4\) grids (Q1178021)

An extended \(4\times 4\) grid is a finite, connected, point, block incidence structure \({\mathcal S}\) such that for any point \(P\) of \({\mathcal S}\), the residue \({\mathcal S}_ p\) (the incidence structure consisting of all points joined to \(P\) and all blocks containing \(P\)) is a \(4\times 4\) grid (which is a generalized quadrangle of order (3,1)). This paper completes the classification of extended \(4\times 4\) grids. For a joined pair of points, \(P\) and \(Q\), let \(\lambda_{P,Q}\) denote the number of points joined to both \(P\) and \(Q\). It is shown that \(\lambda_{P,Q}=6,10\) or 12. If \(\lambda_{P,Q}=6\) for all pairs then \({\mathcal S}\) is a locally \(4\times 4\) grid graph. These structures have been classified by \textit{A. Blokhuis} and \textit{A. E. Brouwer} [J. Graph Theory 13, No. 2, 229-244 (1989; Zbl 0722.05054)]. If \(\lambda_{P,Q}=12\) for one pair then \(\lambda_{P,Q}=12\) for all pairs and \({\mathcal S}\) is the unique 4-uniform extended generalized quadrangle of order (3,1). In the remaining case, there exists a pair for which \(\lambda_{P,Q}=10\), it is shown that \(\mathcal S\) is the unique geometry of 35 points and 56 blocks which is the twisted half \(8\choose 4\) Johnson geometry. A geometric construction of this geometry is provided.