I am confused about what the the following statement means:
"Being $C$ the smallest cyclic code on $\mathbb F_2$ containing the word $w = 110110$".
Maybe it's a stupid question, but does the smallest refer to the dimension or the length?
The exercise also asks me to list $C$'s words and the solution is something like this:
$0$, $w$, $w'$ (first shifting of $w$), and $w+w'$. I would have done all the shifts of $w$, so did he take just one ($w'$) because of the definition of "smallest"?
Being C the smallest cyclic code refer to the dimension(number of word in the code). So the list of C's words, is given by the null vector, w, w' (first cyclic shifts of w), w+w'. At this point we don't need other w's shift given that our set(0, w, w', w+w') is enough for our goal and it's easy to check that this set is a cyclic code.