$(50$ pts$)$ The Hamming weight $($HW$)$ of a binary word is defined as the number of $1s$ that are different from the $0$ also known as the population count or popcount for short. For example, the HW of an $8-$bit binary word $10101100$ is $4($Since$,$ there are four $1$ s in the $8-$bit word$).$