Question

Consider a parity check code with three data bits and four parity check bits. Three of the code words are 0101011, 1001101 and 1110001.

Which of the following are also code words?
I. 0010111 II. 0110110
III. 1011010 IV. 0111010

• A. II and IV
• B. I, II and III
• C. I, II, III and IV
• D. I and III

Answer 1

The correct option is D I and III

$\text{Consider first 3 bits}{x}_1.x_{2}and{x}_{3}are$
data bits and last 4 bits
$c_1,c_2,c_{3}and{c}_4\text{are parity check bits}$
i.e., 0101011 will be
$x_1\left(0\right),x_2\left(1\right),x_3\left(0\right),c_1\left(1\right),c_2\left(0\right),c_3\left(1\right)and{c}_4\left(1\right)$

By analysing pattern used to create parity bits i.e., here

$x_1$	$x_2$	$x_2$	$c_1=x_1⨁x_3$	$c_2=x_2⨁x_3$	$c_3=x_2⨁x_3$	$c_4=x_1⨁x_2⨁x_3$
0 1 1	1 0 1	0 0 1	1 1 0	0 1 0	1 0 0	1 1 1

So by applying same rule to find other code words.

$x_1$	$x_2$	$x_3$	$c_1=x_1⨁x_2$	$c_2=x_1⨁x_3$	$c_3=x_2⨁x_3$	$c_4=x_1⨁x_2⨁x_3$
0 0 1 0	0 1 0 1	1 1 1 1	0 1 $\ne 0$ so fail 1 1	1 1 1 $\ne 0$ so fail	1 1	1 0

$\text{So only}{I}^{st}andII{I}^{rd}\text{are code word.}$