Question

Find the parity-check matrix of each standard generator matrix in Exercise 5 of Section 16.1.

Short answer

1. $\left(\begin{array}{cc}0& 0\\ 1& 1\\ 1& 0\\ 0& 1\end{array}\right)$
   
2. $\left(\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$
   
3. $\left(\begin{array}{c}1\\ 1\\ 0\\ 1\end{array}\right)$
   
4. $\left(\begin{array}{ccc}1& 1& 1\\ 1& 0& 1\\ 1& 1& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$
   

Step-by-step solution

Result used

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

Referred question:

Question 5: List all codewords generated by the standard generator matrix:

1. $\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 1\end{array}\right)$
2. $\left(\begin{array}{ccccc}1& 0& 1& 1& 1\\ 0& 1& 0& 1& 0\end{array}\right)$
3. $\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{array}\right)$
4. $\left(\begin{array}{cccccc}1& 0& 0& 1& 1& 1\\ 0& 1& 0& 1& 0& 1\\ 0& 0& 1& 1& 1& 0\end{array}\right)$

(a)Step 3: Determine parity check matrix for standard generator matrix (a)

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

Order of matrix G: $2\times 4$.

$$\begin{gathered} \mathrm{n}=4 \\ \mathrm{k}=2 \end{gathered}$$

$G=\left(I_2|A\right)$, where $\mathrm{A}=\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)$

So parity matrix is of order $4\times 2$,$\mathrm{H}=\left(\frac{\mathrm{A}}{{\mathrm{I}}_2}\right)$

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

(b)Step 4: Determine parity check matrix for standard generator matrix (b)

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

Order of matrix G: $2\times 5$.

n=5

k=2

$\mathrm{G}=\left({\mathrm{I}}_2|\mathrm{A}\right)$, where $A=\left(\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\end{array}\right)$$H=\left(\frac{A}{I_3}\right)$

So parity matrix is of order $5\times 3$,

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

(c)Step 5: Determine parity check matrix for standard generator matrix (c)

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

Order of matrix G: $3\times 4$.

n=4

k= 3

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

So parity matrix is of order $4\times 1$, $H=\left(\frac{A}{I_1}\right)$

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

(d)Step 6: Determine parity check matrix for standard generator matrix (d)

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

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

n=6

k= 3

$G=\left(I_3|A\right)$, where $A=\left(\begin{array}{ccc}1& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$

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

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