Crooked Functions, Bent Functions, and Distance Regular Graphs

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) 6= 0 for any three distinct x, y, z; Q(x) +Q(y) +Q(z) +Q(x+ a) +Q(y + a) +Q(z + a) 6= 0 if a 6= 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. 1 Crooked functions and bent functions Let V and W be n-dimensional vector spaces over GF(2), and Q : V → W any mapping. We shall use the notation Q(a1, a2, . . . , am) = Q(a1) +Q(a2) + . . .+Q(am). Also, for 0 6= a ∈ V , we denote by Ha(Q), or simply Ha, the set Ha = Ha(Q) = {Q(x) +Q(x+ a) | x ∈ V }. We shall denote the size of a finite set X either by |X| or by #X; whichever notation looks better in the context.