Question

The parity-check matrix of a (5, 2) linear block code (LBC) is given by,

$H=\left[\begin{array}{ccccc}1& 1& 1& 0& 0\\ 0& 1& 0& 1& 0\\ 0& 1& 0& 0& 1\end{array}\right]$

Which of the following is not a valid code word of the given LBC?

• A. [01111]
• B. [11011]
• C. [10100]
• D. [00101]

Answer 1

The correct option is D [00101]

For a code word c,

$c{H}^T=0$

$\text{(a) For}c=\left[10100\right],$

$c{H}^T=\left[10100\right]=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[000\right]$

$\text{(b) For}c=\left[01111\right],$


$c{H}^T=\left[01111\right]=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[000\right]$


$\text{(c) For}c=\left[11011\right],$

$c{H}^T=\left[11011\right]=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[000\right]$

$\text{(d) For}c=\left[00101\right],$

$c{H}^T=\left[00101\right]=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[101\right]\ne 0$

So, the code word given in option (d) is not a valid one.
Note: In the problems related to linear block codes, use EX-OR operation whenever there is an addition operation of bits.