A basic problem in arithmetic geometry is to determine if a hypersurface of degree d over the finite field F_q of q elements has a non-trivial F_q-rational point. When the degree d is very small compared to the field size q (and the hypersurface is smooth), then Deligne's bound (Riemann hypothesis) settles this problem. When the degree d is large, this problem becomes out of reach in general. In this talk, we show that there are many F_q-rational points for complete symmetric hypersurfaces of dimension at least two, even for large degree d. This depends crucially on a coding theory result of Cheng-Li-Zhuang on deep holes of Reed-Solomon codes. This is a joint work with Jun Zhang.