$4.($a$)$ Find the parity check matrix in systematic form for a code over Z$5$capable of correcting single errors. Assume that the number of information bits, k $=8.($b$)$ Find the generator matrix for this code. $($c$)$ Suppose the given information word is $(12041123).$ Find the corre$-$ sponding codeword. $($d$)$ Suppose there is a single error in the fourth information digit which changed from $4$ to $2.$ Explain how error correction is done.