Question

Show that the standard generator matrix

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

generates the$\left(6,5\right)$ parity-check code in Example 2. [Hint: List all the codewords generated by G; then list all the codewords in the parity-check code; compare the two lists.]

Short answer

It has been shown that the given standard generator matrix generates the $\left(6,5\right)$parity check in Example 2.

Step-by-step solution

Given that

Given that Standard generator matrix is

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

Referred question

In Example 2$\left(6,5\right)$ parity-check code is constructed. The message word are the elements of $B\left(5\right)$. A message word is converted to codeword (element of $B\left(6\right)$) by adding a sixth digit to the string. The extra digit is the sum (in${\mathrm{\mathbb{Z}}}_2$ ) of the digits in the message word.

Proof

Let the message word $abcde\in B\left(5\right)$, the corresponding codeword is

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

where $f=a+b+c+d+e$in ${\mathrm{\mathbb{Z}}}_2$(sum of the digits in message word).

This is same as$\left(6,5\right)$ parity-check code was defined in example 2.