$(12$ points$)$ Consider a $10-$bit data, $D=\{d_1,d_2,d_3,d_4,d_5,d_6,d_7,d_8,d_9,d_{10}\},$ to be protected by using a

Hamming single error correcting code with an additional parity bit, $p_{11},$ so that double errors can be

detected.

$($a$)(2$ points$)$ Let the encoded codeword $C=\{p_1,p_2,d_1,p_4,d_2,d_3,d_4,p_8,d_5,d_6,d_7,d_8,d_9,d_{10},p_{11}\}.$

Show the encoding process of $D=1001011001$ to obtain the encoded codeword $C.$

$($b$)(2$ points$)$ Assume no error occurs. Show the decoding process of $C($i$.$e$.,$ locating$/$correcting the

single error, detecting double errors, or confirming that there is no error; you have to verify if the

result is right or wrong$)$ from $($a$).$

$($c$)(2$ points$)$ From $($a$),$ suppose $d_5$ of $C$ is inverted. Show the decoding process.

$($d$)(2$ points$)$ From $($a$),$ suppose $p_1$ and $d_8$ of $C$ are inverted. Show the decoding process.

$($e$)(2$ points$)$ From $($a$),$ suppose $p_1,d_8,$ and $p_{11}$ of $C$ are inverted after the encoding. Show the decoding

process.

$($f$)(2$ points$)$ What is the minimum number of parity bits required to protect $128-$bit data using the same

method to construct a SECDED code?