SEC$-$DED

A communication system uses a SEC$-$DED $($Single Error Correction$-$Double Error Detection$)$ code to protect transmissions. The code is based on a $7-$bit Hamming code extended with an odd parity bit $($the most significant bit $8).$ Using the notation from the notes, each received word consists of the parity bit followed by the $7-$bit Hamming code word $7654321.$ For each of the folloying received words, extract the $4-$bit data word when possible, or mention that the received word had uncorrectable errors.

Hint: Remember that this code is able to correct single errors. If you detect an error and can flip $1$ bit to get to a valid codeword, then you have found and fixed the error. If you can't get to a valid codeword with $1$ bit flip, then the received word is uncorrectable because there may be more than one way to flip two bits to get to valid codewords.

$00101101$

$10011110$

$10101001$

$10010001$

$00111001$