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I need to compute a generator polynomial for a binary cyclic code of length 12 and dimension 5. I know that factorization of $(x^{12}+1)$ over $GF(2)$ is $(x+1)^4(x^2+x+1)^4$. What will be next step?

Thanks for any advice.

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    Is it possible this process? I know that factorization of $(x^{12}+1)$ is $(x+1)^4(x^2+x+1)$ and I need to find the product of factor of degree $n-k$ (in this case 7). So the generator polynomial can be $x^7+x^5+x^4+x^3+x^2+x+1$. Is it true? Can I get a parity check polynomial from this generator polynomial?2012-10-16

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