Generating idempotent (cyclic code length 7)

Question

I have given a cyclic code $C$ generated by $g(x) = x^3 + x + 1$. Now, I'm looking for the generating idempotent of $C$.

Is it correct that I have to find the factorization of $x^3+x+1$ over GF(2), then the minimal generator polynomials and then I'm done?

@JohnD It means that you won't learn anything interesting while trying to factor $x^3+x+1$. — rschwieb, Mar 29 '19 at 13:30

score 1 · Accepted Answer · answered Mar 29 '19 at 14:05

Are you sure you have not covered the theory of how to do this in your text/class? I bet you have, or they wouldn't have asked. But it is understandable to have missed it. I would like to encourage you to go back and pinpoint where it is discussed and how this might be done. One can't learn this way for long - missing important bits and stumbling through the exercises without looking backward for help. (And let me know if the way your book describes it is simpler than my hint below.)

But to give you a hint to your question: a theoretical way to compute the solution is an application of this answer.

If you apply it, you can work out that $(x^3+x+1)\equiv(x^4+x^2+x)\pmod{x^7-1}$ where the latter generator is an idempotent. Be sure to pin down what $A$ and $I\lhd A$ you are talking about before applying it.

Generating idempotent (cyclic code length 7)

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