Crooked Functions, Bent Functions, and Distance Regular Graphs

Abstract

Let $V$ and $W$ be $n$-dimensional vector spaces over $GF(2)$. A mapping $Q:V\rightarrow W$ is called crooked if it satisfies the following three properties: $Q(0)=0$; $Q(x)+Q(y)+Q(z)+Q(x+y+z)\neq 0$ for any three distinct $x,y,z$; $Q(x)+Q(y)+Q(z)+Q(x+a)+Q(y+a)+Q(z+a)\neq 0$ if $a\neq 0$ ($x,y,z$ arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having $\lambda=0$ and $\mu=2$ 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.