ternary cyclic codes of length 27

Question

With $k = 1,2,3,\ldots,26$, is it possible to find a ternary cyclic code of length $27$ and dimension $k$? How can i show that it exists - if it does?

Answer 1

Hint: Over the field $GF(3)$ we have $$ p(x)=x^{27}-1=(x-1)^{27} $$ by the so called freshman's dream (check this if you don't see it right away). Recall that a degree $t$ factor of $p(x)$ generates a code of dimension $k=27-t$.