Now, I would like to show that these kind of evaluations reaches at least 6 classes $\pmod{11}$. For example, in the sequence (1) above one may take $\{b_1,\dots,b_5\} = \{0,0,0,0,7\}$ and it'll just generate 5 elements $\pmod {11}$, namely $7s^4, 7s^3, 7s^2, 7s, 7$. But we instead can take $\{b_1,\dots,b_5\} = \{0,0,0,7,3\}$ and it generates $3s^4 + 7s^3 \equiv 6, 3s^3 + 7s^2 \equiv 7, 3s^2 + 7s \equiv 10, 3s + 7 \equiv 8, 3 + 7s^4 \equiv 2 \pmod{11}$ (5 distinct elements). But it also generates $7s + 3 \equiv 9 \pmod{11}$, so we generate at least 6 distinct elements $\pmod{11}$. (see next..)