Balanced incomplete Block design for testing an experiment I am reading something balanced incomplete block design from a book. I don't understand why is it easy to see that in this design Each vehicle is evaluated 8 times, each test driver evaluates 4 vehicles and every possible pair of vehicles is evaluated by same test driver 3 times ?


To see that each vehicle is evaluated eight times, note that $D_0$ and $D_1$ between them contain eight distinct finite field elements.  Therefore, any given finite field element $z$ occurs in each of the eight blocks
$$
\begin{aligned}
&z-1+D_0 && z-x^2+D_0 && z-x^4+D_0 && z-x^6+D_0\\
&z-x+D_1 && z-x^3+D_1 &&z-x^5+D_1 && z-x^7+D_1,
\end{aligned}
$$
and in no others.
The property that every pair of vehicles is evaluated by the same test driver exactly three times follows from the fact that the pair $D_0$, $D_1$ is a generalized difference set.  Each nonzero field element appears as a difference of two elements of $D_0$ or as a difference of two elements of $D_1$ in exactly three ways, as one can check by computing the $12$ nonzero differences in $D_0$ and the $12$ nonzero differences in $D_1$:
$$
\begin{aligned}
D_0:& \ 2x,1,x+1,x,x+1,2x+1,2,2x+2,x,2x+2,x+2,2x\\
D_1:& \ x+2,x,1,2x+1,1,2x+2,2x,2,2x+1,2,x+1,x+2.
\end{aligned}
$$
Now if two finite field elements $y$ and $z$ appear in a block $w+D_0$ then there are elements $a,b\in D_0$ such that $w+a=y$ and $w+b=z$, which implies that the difference $z-y=(w+b)-(w+a)=b-a$ is a difference of $D_0$.  Similarly, if $y$ and $z$ appear in a block $w+D_1$, then $z-y$ is a difference of $D_1$.
Furthermore, given any finite field elements $y$ and $z$, then any time we have elements $b$ and $a$ of $D_0$ (or of $D_1$) with the same difference, $b-a=z-y$, we have a block $(y-a)+D_0$ (or $(y-a)+D_1$) that contains $y$ and $z$.  In other words, we get a block $w+D_0$ or $w+D_1$ containing $y$ and $z$ for every occurrence of $z-y$ as a difference of $D_0$ or $D_1$.  Since there are three such occurrences, we get three blocks containing $y$ and $z$.
