$($a$)$ Consider the ternary linear code C of length $5$ with parity matrix

H $=$

$$

$$

$02002$

$01221$

$20122$

$$

$$

$($i$)$ What is dim C$?[5$ marks$]$

$($ii$)$ Write out all the elements of C$.[10$ marks$]$

$($iii$)$ Suppose the word y $=20012$ is received. Use coset leader decoding to decode y

to an element of C$.[8$ marks$]$

$($b$)$ Define what is meant by a binary Hamming code. Also, suppose r $>=2$ and suppose

C is a binary Hamming code of length $2$r $1.$ Prove that dim C $=2$r $1$ r$.$