Question

Prove that an element of$B\left(6\right)$is a codeword in the$\left(6, 5\right)$ parity-check code(Example 2) if the sum of its digits is0. [Hint: Compare the sum of the firstfive digits with the sixth digit.]

Short answer

It is proved that an element of $B\left(6\right)$is a codeword in the $\left(6, 5\right)$parity-check code if the sum of its digits is0.

Step-by-step solution

 Step 1: Conclusion from Example 2

If the sum of the message-word digits is 0, a 0is added to makethe codeword; if the sum of the message-word digits 1 is, a 1is added for the codeword$1+1=0$

Proving that an element of B6 is a codeword in the 6, 5parity-check code if the sum of its digits is 0

Suppose X is an element of$B\left(6\right)$ such that sum of digits is 0.

Now we have two cases,

• Sum of the first five digits is 0.

Since the sum of five digits is zero, therefore the sixth digit must be zero. And we know the sixth digit is some of the first five digits.

Hence, we can construct a codeword in the $\left(6,5\right)$parity-check code.

• Sum of the first five digits is 1.

Since the sum of five digits is 1, therefore the sixth digit must be 1 and, and we know the sixth digit is some of five digits and $1+1=0$.

Hence, we can construct a codeword in the $\left(6, 5\right)$parity-check code.

Therefore, in both cases, we can construct a codeword in the$\left(6, 5\right)$ parity-check code. Hence it is proven that an element of$B\left(6\right)$ is a codeword in the$\left(6, 5\right)$ parity-check code if the sum of its digits is 0.