$4.$ Consider error detection codes with the following parameters: $(15$pt$)$

$\backslash ($ m $\backslash )$ : number of message bits

$\backslash ($ r $\backslash )$ : number of redundancy bits

$\backslash ($ n $\backslash )$ : codeword length $\backslash (($n$=$m$+$r$)\backslash )$

$\backslash ($ k $\backslash )$ : maximum number of bit errors to be corrected

$($a$)$ Establish a relationship between $\backslash ($ m $\backslash )$ and $\backslash ($ r $\backslash )$ for $\backslash ($ k$=2\backslash ).$

$($b$)$ Establish a relationship between $\backslash ($ m $\backslash )$ and $\backslash ($ r $\backslash )$ for an arbitrary $\backslash ($ k $\backslash )$ value such that $\backslash ($ k