Google scholar gives more than 200 articles comcerning finite Bolyai-Lobachevsky (BL) planes. Usually they devoted to construction of such objects (axioms may be different).
Are their any applications of these objects in combinatorics, coding theory etc?
I know one example: Kirkman's schoolgirl problem. But it is slightly artificial because it can be solved using projective geometry of three dimensions over $GF(2)$.
The following axioms are taken from The Existence of Finite Bolyai-Lobachevsky Planes by Steven H. Heath.
Axiom 1. If $P$ and $Q$ are two points there is exactly one line containing $P$ and $Q$.
Axiom 2. If $l$ is any line there is a point $P$ which does not lie on $l$.
Axiom 3. There are at least two points on every line.
Axiom 4. There exists at least one line.
Axiom 5. Given a point $P$ not on a line $l$, there are exactly $k$ lines which are parallel to $l$ and pass through $P$.
A finite BL space satisfying Axioms 1-5 will be denoted by $BL(k, n)$.
For example $3$-dimensional finite projective geometry with $3$ points on each line is $BL(4,3)$. It leads to a solution of the Kirkman's Fifteen School Girl Problem. And any solution of this problem is a $BL (4, 3)$.
