I just copy the notation quickly. A variable with a space and then a number it means is a power.
I have BCH cyclic code like that:
B = BCH(7) of length 15 = 2 4 − 1. F 16 = F 2 [x]/(x 4 + x + 1), α ∈ F 16 is primitive. Then α, α 2 , α 3 , α 4 , α 5 , α 6 are among the roots of B. The cyclotomic classes of 2 mod 15 that we need are C 1 = {1, 2, 4, 8}, C 3 = {3, 6, 9, 12} and C 5 = {5, 10}. So g(x) = f C 1 (x)f C 3 (x)f C 5 (x), its degree is 10, and k = dim B = 5. Assume we receive the word: y = 111011100000110 = 1 + x + x 2 + x 4 + x 5 + x 6 + x 12 + x 13
I hope you can understand it. Till here is everythink ok. But later it says:
s0 = y(α) = 1 + α + α 2 + α 4 + α 5 + α 6 + α 12 + α 13 = α 6
s1 = y(α 2 ) = y(α) 2 = α 12
s2 = y(α 3 ) = 1 + α 3 + α 6 + α 12 + α 15 + α 18 + α 36 + α 39 = α 8
The problem is, how it computes s0? Or s2? Why s0 = α power 6?
Best Answer
Hint: Since $\alpha$ is a zero of $x^4+x+1$, we have $\alpha^4+\alpha+1=0$ and so $\alpha^4=\alpha+1$. Now you can compute $\alpha^5=\alpha(\alpha+1)=\alpha^2+\alpha$ and so on. This allows to represent each power $\alpha^k$ in the basis $\{1,\alpha,\alpha^2,\alpha^3\}$. Then its clear how to compute the syndrome.