Question

Find the parity-check matrix for the code in Example 7 of Section 16.1

Short answer

Parity check matrix is $\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

Step-by-step solution

Result used

Consider an $\left(n,k\right)$code with $k\times n$standard generator matrix data-custom-editor="chemistry" $\mathrm{G}=\left({\mathrm{I}}_{\mathrm{k}}|\mathrm{A}\right)$. The parity-check matrix of the code is the $\mathrm{n}\times \left(\mathrm{n}-\mathrm{k}\right)$ matrix $H=\left(\frac{A}{I_{n-k}}\right)$.

Referred question

Example 7 of Exercise 16.1 :

The standard generator matrix is given as:

$\mathrm{G}=\left(\begin{array}{cccccc}1& 0& 0& 0& 1& 1\\ 0& 1& 0& 1& 0& 1\\ 0& 0& 1& 1& 1& 0\end{array}\right)$

The code is $\left(6,3\right)$.

Determine parity check matrix

Standard generator matrix: $G=\left(\begin{array}{cccccc}1& 0& 0& 0& 1& 1\\ 0& 1& 0& 1& 0& 1\\ 0& 0& 1& 1& 1& 0\end{array}\right)$.

Order of matrix G: $3\times 6$.

n=6

k=3

$\mathrm{G}=\left({\mathrm{I}}_3|\mathrm{A}\right)$, where data-custom-editor="chemistry" $\mathrm{A}=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$

So, parity matrix is of order$6\times 3$ , $H=\left(\frac{A}{I_3}\right)$

Thus, Parity matrix$H=\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$