Modulo a prime p, for what values of x is a cubic polynomial in x a quadratic residue (i.e. a square). Hasse's Theorem says that for about p/2 (plus-minus sqrt(p)) values of x, the polynomial is a quadratic residue. Hasse was one of the first to work on elliptic curves.
Andrew Weil proved an even more general theorem, for arbitrary degree polynomials. Consider a code, in which for each poynomial we have a code word, whose xth bit is 1 iff the polynomial(x) is a quadratic residue. Then all codewords are almost of the same weight, and you can show that their distances from each other is high-- giving you nice codes.
Ref: This was worked out in "Simple Constructions of Almost k-wise Independent Random Variables" by Alon,Goldreich,Hastad, Peralta in STOC 90.