Weight divisibility of cyclic codes, highly nonlinear functions on F(2)m, and crosscorrelation of maximum-length sequences

We study [2(m) - 1, 2m]-binary linear codes whose weights lie between w(0) and 2(m) - w(0), where w(0) takes the highest possible value. Primitive cyc lic codes with two zeros whose dual satisfies this property actually corres pond to almost bent power functions and to pairs of maximum-length sequence s with preferred crosscorrelation. We prove that, for odd m, these codes ar e completely characterized by their dual distance and by their weight divis ibility. Using McEliece's theorem we give some general results on the weigh t divisibility of duals of cyclic codes with two zeros; specifically, we ex hibit some infinite families of pairs of maximum-length sequences which are not preferred.