Let C be the linear $[5,3]-$code over F$3$ with check matrix

$10201$

$21021!$

$.$

Determine the minimum distance of C$.[6$ marks$]$

$($b$)($i$)$ What is the definition of the Hamming code Ham$($r$,$ q$)?[4$ marks$]$

$($ii$)$ Construct a parity check matrix for the Hamming code Ham$(2,7).[4$ marks$]$

$($iii$)$ Explain how a single error can be corrected in Ham$(2,7).$ Decode the word $111111111$