A geometric construction

In an n-dimensional projective geometry, a correlation is a 1-1 mapping of points to planes. In this mapping, the meet of two points (a line) is mapped onto the meet of two planes, and so forth. If the correlation is of order 2, then it is called a polarity.

When n is 3, the meet of 2 planes is a line, and so lines are mapped onto lines. For the rest of this explanation, we shall assume n is 3, although the same principle applies for n any odd number.

When the projective geometry is finite, say PG(n,q), where q is the order of a Galois Field (i.e. q is a prime or a power of a prime), the lines can be interpreted as the blocks of a BIBD, and the planes can be interpreted as the blocks of a SBIBD.

Moreover, taking a given point, and the set of lines in which it occurs, the lines with which it is matched comprise the set of lines in the plane with which it is matched. For, suppose the point is denoted by A, and its corresponding plane by α. The line AB is matched with the line αβ,in an obvious notation: if the point C also lies on AB, then the plane γ must intersect with (contain) the line αβ. So, to the set of distinct lines AB, AD, etc, corresponds the set of distinct lines αβ, αδ, etc, namely all the distinct intersections of planes with α, which is the set of all the lines in α.

Moreover, if there are (q+1) points on the line AB, then there are (q+1) planes through the line αβ. And, the points with which the typical point A occurs, are just the points of the plane with which it corresponds, each (q+1) times (by simple counting).

To relate this to the OBIBD construction, let n10 be the incidence matrix of the original points to lines, and let n20 be the matrix of the points of the corresponding lines, taken in the same order. Then the above exposition says that n20n10' is (q+1) times the incidence matrix of a SBIBD. To get an OBIBD, it will be necessary to match the points in the corresponding lines so that n21 is the incidence matrix of the same SBIBD.

Different sorts of polarity exist: in geometric terms, the most obvious questions of interest concern how many points lie in the corresponding planes: this is not considered further, though might be of some interest to design theorists in this context, too.

An example is given here

To complete the construction, see here.