I have a question that involves showing there do not exist vectors $v_1,v_2,v_3 \in \mathbb{Z}_2^6$ such that:
$wt(v_i)\geq4$ for $i \in \{1,2,3\}$
$wt(v_i+v_j)\geq3$ for $i \in \{1,2,3\}$
$wt(v_1+v_2+v_3)\geq2$
where $\mathbb{Z}_2^6$ is the set of binary words of length 6 ($\mathbb{Z}_2$ can be considered as the set of residue classes mod 2, and addition as residue class addition).
and $wt(v)$ is the weight function (the number of 1's in the vector v).
I originally tried finding vectors in $\mathbb{Z}_2^6$ that did satisfy the three conditions above, and couldn't, only to give up and look at the mark scheme, which mentions that the above should be simple to show.
Any hint would be appreciated.
Thanks in advance :)