$($a$)$ Suppose we are using the Reed Muller code $RM(3)$ and $u=01111010$ is received.

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

to decode $u$ to an element of $RM(3).$

$[7$ marks$]$

$($b$)$ Suppose $C$ is a linear code with length $7.$ Suppose $.$ Prove that

dimC$3.$

$[6$ marks$]$

$($c$)$ Consider the binary code $C=\{1011101,0110001,1101100,0010111\}.$ Find the

greatest number of errors that $C$ can correct.

$[6$ marks$]$

$($d$)$ Prove there is no ternary code $C$ with $|C|=23$ such that $C$ has length $7$ and

corrects $2$ errors.

$[8$ marks$]$

$($e$)$ Suppose $C$ is a linear binary code with basis $\{100,010\}.$ Find all the elements in

$.$

$[8$ marks$]$