$3.$ Suppose we want an error$-$correcting code that will allow all single$-$bit errors to be corrected for memory words of length $10.$

a$.$ How many parity bits are necessary?

b$.$ Assuming we are using the Hamming algorithm presented in this chapter to design our error$-$correcting code and an odd parity $,$ find the code word to present the $10-$bit information word:

$1001100111$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

c$.$ We now receive the following code word:

$10101100001011$

Is this a legal code word, assuming an odd parity? If not, what is the corrected code word? $($Which bit is the error bit?$)$