## Vector whose coordinates are pairwise independent

Consider the following twelve vectors in $\lbrace 0,1 \rbrace^4$ :

$\begin{array}{l} V_1=(0,0,0,0),V_2=(0,0,0,1),V_3=(0,0,0,0), \\ V_4=(0,1,0,1),V_5=(0,1,1,0),V_6=(0,1,1,1), \\ V_7=(1,0,0,1),V_8=(1,0,1,0),V_9=(1,0,1,1), \\ V_{10}=(1,1,0,0),V_{11}=(1,1,0,0),V_{12}=(1,1,1,1). \\ \end{array}$

Then the multiset $V=\lbrace V_k\rbrace_{1 \leq k \leq 12}$ (note that $V_{11}=V_{10}$) has the following property : if $\overrightarrow{X}$ is a random vector whose distribution is unform in $V$, then the coordinates of $\overrightarrow{X}$ are pairwise independent. Moreover, $V$ is minimal with respect to this property. Can $V$ be “explained” rather than just enumerated?