I am puzzled about the following question:

Let C be a smooth, projective, absolutely irreducible curve defined over GF(q) and let g denote the genus of C. O is a rational point on C, and the divisor D is defined as D = (2g-1)O.

Question: Whether there exist rational points $P_1,P_2,\cdots,P_n~(n>g)$ on C such that for any g rational points $P_{i_1},P_{i_2},\cdots,P_{i_g}$ of them, $l(D-\sum^g_{j=1}P_{i_j})=0$ (i.e. the dimension of $L(D-\sum^g_{j=1}P_{i_j})$).

I know it is easy to find g such points. But the existence of further points really confused me.

Thank you for your help!