Let Ci be an $[$n$,$ ki$,$ di$]-$linear code over Fq $,$ for i $=1,2.$ Define

C :$=\{($a $+$ x$,$ b $+$ x$,$ a $+$ b $+$ x$)$ : a$,$ b in C$1,$ x in C$2\}.$

$($a$)$ Show that C is a $[3$n$,2$k$1+$ k$2]-$linear code over Fq $.$

$($b$)$ If Gi is a generator matrix of Ci $,$ for i $=1,2,$ find a generator matrix

of C in terms of G$1$ and G$2.$

$($c$)$ If Hi is a parity$-$check matrix of Ci $,$ for i $=1,2,$ find a parity$-$check

matrix of C in terms of H$1$ and H$2.$