I have a coding theory assignment and I thought it would be a good idea to double check before I hand it in. I'm asked to find the idempotent generators of the four binary QR codes C1, C2, C3, C4, of length 7. Here's what I came up with:
The cyclotomic cosets would be: {0}, {1, 2, 4}, {3, 6, 5}, and the idempotents in $R_7$ are $$ \begin{aligned} &0, 1 = x^0,\\ &x+x^2+x^4, \\ &x^3+x^6+x^5,\\ &1+x+x^2+x^4,\\ &1+x^3+x^6+x^5,\\ &x+x^2+x^4+x^3+x^6+x^5, \\ \text{and}\qquad &\\ & 1 +x+x^2+x^4+x^3+x^6+x^5. \end{aligned} $$