$($a$)$ Prove there is no binary code C with $|$C$|=20$ such that C has length $12$ and

corrects $3$ errors. $[9$ marks$]$

$($b$)$ Consider the ternary code C $=\{1111111,0202011,2020211,0011022\}.$ Find the

greatest number of errors that C can correct. $[7$ marks$]$

$($c$)$ Suppose we are using the Reed Muller code RM$(3)$ and u $=00011001$ is received.

Suppose exactly one error occurred during transmission. Use majority logic decoding

to decode u to an element of RM$(3).[10$ marks$]$

$($d$)$ Is there a ternary code C with $|$C$|=6$ such that C has length $9$ and corrects $2$

errors? Justify your answer.