Bounding OR-type coordinates between $n$ Boolean vectors in a ball of radius $r$ Say I have set of $m$ Boolean vectors 
$$B = \{x_1,\ldots, x_m\}$$ 
where $x_i  = [x_{i,1},\ldots, x_{i,n}] \in \{0,1\}^n$ for all $x_i \in B$.
I know the following about the vectors $x_i \in B$:
(i) $|x_i| \in [1,n-1]$ for all $x_i \in B$ (at least 1 zero coordinate, and one non-zero coordinate)
(ii) $d(x_i, x_j) \geq 1$ for all $x_i, x_j \in B$ (all vectors are distinct)
(iii) $d(x_i, x_j) \leq r$ for all $x_i, x_j \in B$ (*all vectors differ by at most $r$ coordinates)
Here, $d(x_i, x_j) = |x_i - x_j| = \sum_{k=1}^{k=n} 1[x_{i,k} \neq x_{j,k}]$.
My question is the following: Given a vector $x_i \in B$, what is maximum number of coordinates where $x_i$ is zero, but at least one of the other vectors in $B$ is non-zero? That is, what is the largest possible size of the following set:
$$Z(x_i) =  \left\{k =1,\ldots, n ~\Big|~ x_{i,k} = 0 ~\text{and}~ \sum_{j \neq i} x_{j,k} \geq 1 \right\} $$

Partial answer:
I can derive a non-trivial bound using only (iii) as shown below. I am not sure if this bound is tight, or whether it can be improved. In particular, my bound does not account for (i) and (ii). 
To get the bound, first define: 
$$S_{01}(x_i,x_j) = \{k ~|~ x_{i,k} = 0, x_{i,j} = 1\}$$
and note that: $$|S_{01}(x_i,x_j)| = \frac{d(x_i,x_j) + |x_j| - |x_i|}{2}$$
Since $\sum_{j \neq i} x_{j,k} \geq 1$ requires that $x_{j,k} = 1$ for at least one $j \neq i$, we have the following upper bound: $$\begin{align}
|Z(x_i)| &\leq \max_{j \neq i} |S_{01}(x_i,x_j)| \\ 
&=\max_{j \neq i} \frac{d(x_i,x_j) + |x_j| - |x_i|}{2} \\ 
&\leq \frac{r - |x_i| + \max_{j \neq i}  |x_j|}{2} 
\end{align}$$

If I didn't misunderstand the question, the following set is a counterexample to your bound with $r=4,|x_1|=1,|x_j|=2,2\leq j\leq 7$ and $Z(x_1)=4.$
$$
\begin{array}{lr}
x_1 & 10000\\
x_2 & 01100\\
x_3 & 01010\\
x_4 & 01001\\
x_5 & 00110\\
x_6 & 00101\\
x_7 & 00011
\end{array}
$$
