Support of a codeword in a binary linear code proof Let $C$ be the binary linear code with the following generator matrix

$G= \begin{bmatrix}
    1 &amp; 1 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 \\
    0 &amp; 1 &amp; 1 &amp; 0 &amp; 1 &amp; 0 &amp; 0  \\
    0 &amp; 0 &amp; 1 &amp; 1 &amp; 0 &amp; 1 &amp; 0 \\
0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 0 &amp; 1 
\end{bmatrix}$

The support of a codeword is the set of coordinate positions in which the code word has nonzero entries. Let $w$ $\in C$ be a code word of weight $4$. Show that there is no code word of weight $3$ in $C$ whose support is a subset of the support of $w$.

I think I have worked out that the codewords of weight $4$ in $C$ are: $\{(1001011),(0010111),(0111001),(0101110),(1100101),(1110010),(1011100)\}$ so does this mean that the support is $\{(1,4,6,7),(3,5,6,7),(2,3,4,7)$ and so on..$\}$?
Then the codewords of weight $3$ are $\{(1101000),(0110100),(0011010),(0001101),(1000110),(1010001),(010001)\}$ ? with support of  $\{(1,2,4),(2,3,5),(3,4,5)$ and so on..$\}$?

I'm not sure if this is right so far? But this is as far as I can get and do not know what to do next to show that there is no code word of weight $3$ in $C$ whose support is a subset of the support of $w$.
If a codeword of weight 3 existed whose support was a subset of a codeword of weight 4, then you must have a codeword of weight 1 (since the difference of codewords is also a code word).

A codeword of weight 1 has one 1 and the rest zeros. Now, can you show that these don't exist?