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First we consider some examples.   \begin{Example}For a given set vector  of bits $S$, for example $S=\(x,y,z\)$, $S=\(x,y,z\)\in \Fb^3$,  we define the "parity bit" a new bit that is $1$ if the number of bits $1$ in $S$  is odd, $0$ otherwise. To compute the parity bit we define a function $f$, $f$  onthe bits in  $S$, thatinput of the function that gives as output the   . For example if $S$ has $3$ bits, that  is $S=\{x,y,z\}$, we have if  the function $f:(\Fb)^3 bits are $n=3$   \[  f:(\Fb)^3  \rightarrow \Fb$, that is $f(x,y,z)=x+y+z$ \Fb,  \]  where $x+y+z\in \Fb$. $f(x,y,z)=x+y+z$.  \end{Example}  Verify that the new bit $f(x,y,z)$ is the parity bit.