Self-dual code from parity-check matrix

Question

I am trying too make a self-dual code from this parity-check matrix:
_ _ | 1 1 1 1 1 1 1 1 |
| 1 1 1 1 0 0 0 0 |
H= | 1 1 0 0 1 1 0 0 |
|1 0 1 0 1 0 1 0|

Thanks for any help!

Answer 1

The code $C$ with parity-check matrix $H$ is indeed a self-dual code. You can prove it by showing that $C$ has $H$ as a generator matrix.

More detailed hint: (hover to view)

Use elementary row operations to put $H$ in the form $[P | I]$ (where $I$ is the identity matrix). Then the generator matrix of $C$ is given by $G = [I | -P^T]$. Observe that the reduced row echelon forms of $G$ and $H$ are the same. Some more details on Wikipedia.