Uniformly packed codes and more distance regular graphs from crooked functions

Let V and W be n-dimensional vector spaces over GF(2). A function Q : V --> W is called crooked (a notion introduced by Bending and Fon-Der-Flaass) if it satisfies the following three properties: Q(0)=0; Q(x)+Q(y)+Q(z)+Q(x+y+z) not equal 0 for any three distinct x,y,z; Q(x)+Q(y)+Q(z)+Q(x+a)+Q(y+a)+Q(z+a)not equal0 if a not equal 0 (x,y,z arbibrary). We show that crooked functions can be used to construct distance regular graphs with parameters of a Kasami distance regular graph, symmetric 5-class association schemes similar to those recently constructed by de Caen and van Dam from Kasami graphs, and uniformly packed codes with the same parameters as the double error-correcting BCH codes and Preparata codes.