I'm struggling since a while about understanding Curtis' MOG. To land there, I'm going through Cullinane's seminal square representation of $PG(3,2)$, as depicted here:
http://finitegeometry.org/sc/pg/dt/ortholatin.html
It's not clear to me how those square matrices are gotten. I mean, which coordinates you have to assign to the $15$ $PG(3,2)$ points to got those lines and planes? Are the "natural" binary $15$ first numbers enough to achieve that?
A similar explanation can be grasped from wikipedia, on $PG(3,2)$ square representation explanation:
A $3-(16,4,1)$ block design has $140$ blocks of size $4$ on $16$ points, such that each triplet of points is covered exactly once. Pick any single point, take only the $35$ blocks containing that point, and delete that point. The $35$ blocks of size $3$ that remain comprise a $PG(3,2)$ on the $15$ remaining points. If these $16$ points are arranged in a $4 \times 4$ grid and assigned $4-\text{bit}$ binary coordinates like in a Karnaugh map for example, one gets the square representation. Geometrically, the $35$ lines are represented as a bijection with the $35$ ways to partition a $4 \times 4$ grid into $4$ regions of $4$ cells each, if the grid represents an affine space and the regions are $4$ parallel planes.
Any help on how this construction has been achieved, is highly appreciated. Thanks
