I am asked to decode this word but the concepts I've learnt tell me that the question can't be done.
Let C be a binary cyclic code generated by $$g(x) = 1 + x + x^8 + x^{12} + x^{14}$$ in F_2[x]/($x^{15}-1$). Assuming no more than 2 errors have occurred, decode $1 + x + x^2 + x^3 + x^6 + x^8 + x^{11} + x^{12} + x^{14}$.
Firstly, g(x) does not divide $x^{15}-1$, so g(x) does not generate a cyclic code. Secondly, even if it does, the dimension of the code would be 1, meaning there are 2 words in the code: Zero and g(x) itself.
Please correct me if my basic concepts are not right!:)
Yes, you're right on both accounts. In the context of cyclic codes it is standard that the generator polynomial is a factor of $x^L-1$, where $L$ is the length of the code. Something is wrong.
Algebraically $g(x)$ and $\gcd(g(x), x^{15}-1)$ generate the same cyclic code (= an ideal of the ring $F_2[x]/(x^{15}-1)$), so one might think that you are first expected to compute the greatest common divisor. But I would be very hesitant in calling $g(x)$ a generator polynomial in that case (e.g. the standard encoding algorithms cannot use this $g(x)$), so I very much doubt that that's what was meant.
Mind you, $\gcd(g(x), x^{15}-1)=(1+x+x^4)(1+x+x^2+x^3+x^4)$ is the generator polynomial of a double-error-correcting BCH-code of length 15. This fits in a way, but I still think that the problem set up is misguided.