I want to construct a $(3v - 2,3,1)$-BIBD from a $(v,3,1)$-BIBD and a quasigroup of order $v-1$.
Attempt: For new BIBD, firstly, I need $3v-2$ points. Let $A$ be a block in $(3v - 2,3,1)$-BIBD and $\left( {Y, \circ } \right)$ be a quasigroup of order $v-1$. I define the points as ${Y_1} = \left\{ {Y \times A} \right\} \cup \infty $. Then, $\left| {{Y_1}} \right| = 3v - 2$. Now, I need to define the blocks for new BIBD. But it seems that there is a problem here because, by such a definition for points, I don't use $(3v - 2,3,1)$-BIBD.
That is, I'm on stuck. I'll be pleasure if one gives a hint for solution.