Crooked functions, bent functions, and distance regular graphs

T. D. Bending ; D. Fon-Der-Flaass

The electronic journal of combinatorics, Tome 5 (1998)

Résumé

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.

Détail

DOI : 10.37236/1372

Classification : 05E30, 05B20, 11T71
Mots-clés : crooked function, distance regular graph, bent functions