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I try to show that the following defines an association scheme: Let $\mathbb{F}_q$ be a field, $\omega$ a primitive element of $\mathbb{F}_q^\times$ and $s$ divides $q-1$. Define $r=\frac{q-1}{s}$, $C_0=\{0\}$, $C_1=\langle \omega^s\rangle=\{\omega^{sk}:k=0,1,\ldots,r-1\}$ and$$ C_i=\omega^{i-1}C_1\quad\text{ for }i=2,\ldots,s. $$I have to show that the $C_i$ partition the set $\mathbb{F}_q$. I tried to construct an group operations which has the $C_i$ as its orbits, but so far I did not succeed. Does anyone have an idea?

Sincerely, Hypertrooper

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  • $\begingroup$ See the answer below. $\endgroup$ – Hypertrooper Jun 11 '18 at 16:10
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Sometimes, it is to difficult to see the easiest solution. The $C_i$ are the equivalence classes of the quotient $\mathbb{F}_q^*/C_1$.

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