We suggest a new method for coloring generalized Kneser graphs based on hypergraphs with high discrepancy and a small number of edges. The main result provides a proper coloring of K(n,n∕2−t,s) in (4+o(1))(s+t) ² colors, which is produced by Hadamard matrices. Also, we show that for colorings by independent set of a natural type, this result is the best possible up to a multiplicative constant. Our method extends to Kneser hypergraphs as well.