Chapter 2, Problem 68E

Explanation of Solution

Code word representation using hamming algorithm:

The length of memory word is 7.

The number of parity bits for memory word can be calculated using the following formula,

$\text{m}\text{+}\text{r}\text{+}\text{1}\le {\text{2}}^{\text{r}}$

Substitute, “7” for “m” in the above formula,

$\begin{array}{l}7\text{+}\text{r}\text{+}\text{1}\le {\text{2}}^{\text{r}}\\ 8\text{+}\text{r}\le {\text{2}}^{\text{r}}\end{array}$

Hence, r must be greater than or equal to 4.

Thus, the number of parity bits required for 7 length memory word is 4.

To create the code for memory word with 7-bit length, 4 parity bits are added compulsorily.

The number of bits present in the code word can be calculated using the following formula,

$\text{n}\text{=}\text{m}\text{+}\text{r}$

Substitute, “7” for “m” and “4” for “r” in the above formula,

$\begin{array}{c}\text{n}\text{=}7\text{+}4\\ \text{=}\text{11}\end{array}$

Thus, the number of bits present in the code word is 11.

The number bits starts from right to left with 1.

The number equal to the power of 2 is known as parity bit.

For example: $2^0=1$, $2^1=2$, $2^2=4$ and $2^3=8$

The positions of the parity bits are 1, 2, 4 and 8.

Now, write the sum of the numbers with powers of 2.