$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 11.2}$

*Computing a parity check matrix for a code specified by a generator matrix.*

a) Show that adding a column of a generator matrix $G$ for a code $C$ to another column produces an alternative generator matrix $G'$ for the same code $C$.

b) Show that for any $[n,k]$ code there is a generator matrix of the form $\left(\begin{array}{c}A\\\hline I\end{array}\right)$ where $A$ is a $(n-k)\times k$ matrix and $I$ is the $k\times k$ identity matrix.

c) Show that if $G = \left(\begin{array}{c}A\\\hline I\end{array}\right)$ then the $(n-k)\times k$ matrix $P =(I|A)$, where $I$ is the $(n-k)\times(n-k)$ identity matrix, is a parity check matrix for the code $C$.

d) Show that if a parity check matrix $P'$ has the form $(A|I)$, then $G' = \left(\begin{array}{c}I\\\hline A\end{array}\right)$ is a generator matrix for the code.