$7\mathrm{.}27$ An error pattern $e$ of a code $C$ is said to be detectable if $e+c!$inC for all cinC. Show that, for a cyclic code, if an error pattern $e(x)$ is detectable, then its $i$ th cyclic shift is also detectable, for any $i.$

$7\mathrm{.}28$ Let $C$ be a binary $7,4-$Hamming code with generator polynomial $g(x)=1+x+x^3.$ Suppose each of the following received words has at most one error. Decode these words using error trapping:

$($a$)1101011$ ;

$($b$)0101111$;

$($c$)0100011.$