My goal is to find a generator polynomial for a binary $[15,5,d]$ cyclic code that corrects all errors with weight less than or equal to $3$.
I have to find a degree $10$ monic polynomial that divides $x^{15}-1$ ($\bmod 2$) and has weight $7$. If $15$ and $5$ were coprime I would know how to factor $x^{15}-1$ using cyclotomic theory. But that is not the case.
Is my reasoning ok? How do I factor $x^{15}-1\bmod 2$? Thanks in advance!
Best Answer
If you try systematically dividing by quartics you will see that more polynomials than $x^4+x^3+x^2+x+1$ factor into $x^{15}+1$ over $\mathbb F_2$: $$x^{15}+1=(x+1)(x^2+x+1)(x^4+x^3+x^2+x+1)(x^4+x^3+1)(x^4+x+1)$$ Then $$(x^2+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)=x^{10}+x^9+x^8+x^6+x^5+x^2+1$$ is the desired weight-$7$ polynomial.