Question

Find the parity-check matrix for the parity-check code in Example 2 of Section 16.1. [See Exercise 7 in Section 16.1.]

Short answer

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

Step-by-step solution

Result used

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

Referred question

As per Example 2 and exercise 7 of Exercise 16.1 :

The standard generator matrix is given as:

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

The code is $\left(6,5\right)$.

Determine parity check matrix

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

Order of matrix G: role="math" localid="1658921649559" $5\times 6$

n=6

k=5

$G=\left(I_5|A\right)$, where $A=\left(\begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\end{array}\right)$

So, parity matrix is of order $6\times 1$, $H=\left(\frac{A}{I_1}\right)$

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