Let C be a linear code over F $=$ GF$(2)$ with a parity$-$check matrix

H $=$

$$

$$

$$

$$

$$

$$

$$

$110000000$

$101000000$

$000110000$

$000101000$

$000000110$

$000000101$

$$

$$

$$

$$

$$

$$

$$

$1.$ What are the parameters n$,$ k$,$ and d of C $?$

$2.$ Write a generator matrix of C$.$

$3.$ Find the largest integer t such that every pattern of up to t errors will be

decoded correctly by a nearest$-$codeword decoder for C$.$

$4.$ A codeword of C is transmitted through an additive channel $($ F$,$ F $,$ Prob$)$

and the word

y $=101010101$

is received. What will be the respective output of a nearest$-$codeword

decoder for C when applied to y$?$

$5.$ Given that the answer to part $4$ is the correct codeword, how many errors

does a nearest$-$codeword decoder correct in this case? And how is this

number consistent with the value t in part $3?$