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?

 

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