Crooked functions, bent functions, and distance regular graphs

Abstract

Let V and W be n-dimensional vector spaces over GF(2). A mapping Q : V → W is called crooked if it satisfies the following three properties: Q(0) = 0; Q(x) + Q(y) + Q(z) + Q(x + y + z) ≠ 0 for any three distinct x, y, z; Q(x) + Q(y) + Q(z) + Q(x + a) + Q(y + a) + Q(z + a) ≠ 0 if a ≠ 0 (x, y, z arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having λ = 0 and μ = 2 which is a cover of the complete graph. Our approach is a generalization of a recent construction found by de Caen, Mathon, and Moorhouse. We study graph-theoretical properties of the resulting graphs, including their automorphisms. Also we demonstrate a connection between crooked functions and bent functions.