I've been reading about Golay codes, and I found an interested property that I am having trouble showing. The property says:
If the $[24, 12, 8]$ binary Golay code $\mathcal{G}$ is punctured in any coordinate and the resulting code is extended in the same position, then the exact same code $\mathcal{G}$ is obtained.
Can anyone offer any suggestions? Thanks in advance!
EDIT
Puncturing and extending are defined as follows:
If $\mathcal{C}$ is an $[n,k,d]$ code over $\mathbb{F}_{q}$, then $\mathcal{C}$ is punctured by deleting the same coordinate $i$ in each codeword. If $G$ is a generator matrix for $\mathcal{C}$, then a generator matrix for the punctured code is obtained by $G$ by deleting column $i$ and omitting a zero or duplicate row that may occur.
The extended code $\mathcal{C}_{ext}$ is the code: $$ \mathcal{C}_{ext} = \{ x_1 x_2 \dots x_{n+1} \in \mathbb{F}_{q}^{n+1} : x_1x_2\dots x_n \in \mathcal{C} \mbox{ with } x_1 + x_2 + \dots + x_{n+1} = 0 \}.$$ The generator matrix for the extended code is obtained from $G$ by adding an extra column to $G$ so that the sum of the coordinates of each row is 0