The telephone company, LEMON, utilizes a specific error correction technique called a "bestof$-$five" code for some of their digital data channels. In this system, each data bit is transmitted

five times. At the receiving end, a majority vote is used to determine the final value of each

data bit. Suppose the uncoded bit error probability in the channel is $10-3$

$.$

a$.$ Calculate the bit$-$error probability at the decoder when using the "best$-$of$-$five" coding

scheme.

b$.$ For a different data transmission scenario, the minimum distance requirement $($dmin$)$

for a linear block code is $11.$ Find the maximum error$-$correcting capability, the

maximum error$-$detecting capability, and the maximum erasure$-$correcting capability.

c$.$ In a backup system, you need to set up a $(7,4)$ Hamming code with the following

generator matrix $($G$)$: