Q$4[10$M$][$Attempt two of the following$]$

$($A$)$Assume a binary repetition code with codeword length of $5.$ If the uncoded bit error probability $p$ is

${10}^{-3},$ calculate the coded bit error probability when using this best$-$of$-$five code.

$($B$)$ Prove the Singleton bound, that is$,$ the minimum distance for an $(n,k)$ linear code is bounded by

$d_{\text{m}\text{i}\text{n}}n-k+1.$ What we called the code for which the equality holds. $?$

$($C$)$ Consider a double$-$error$-$correcting code for $9$ data bits. How many check bits are required?