$($Noisy communication$)$ Let $>0$ be a small constant. Consider communication in a very noisy

environment in which a sent bit $0$ will be received as $0$ with probability $1$

$2+$ and will be received as

$1$ with probability $1$

$2.$ Similarly for a sent bit $1($it will be received as $1$ with probability $1$

$2+$ and

will be received as $0$ with probability $1$

$2).$ To encode a single bit b$,$ one can use a repetition code

of length n in which the n length word consisting of repeating bit b n$-$times is sent. At the decoder, if

more $0$s than $1$s were received, the decoded bit $$b is set to $0,$ and otherwise to $1.$ Communication is

successful if the message bit b equals the decoded bit $$ b$.$ Use the bounds presented in class to compute

a lower bound for n $($as a function of $)$ such that the decoding error will be at most e$100.$ That is$,$

specify how large n needs to be in order to obtain a decoding error less that e$100.$ Prove your answer

in detail. Specify in detail which bounds were used.