language: en

Crooked Functions, Bent Functions, and Distance Regular Graphs

T.D. Bending D. Fon-Der-Flaass
School of Mathematical Sciences, Queen Mary and Westfield College,
London E1 4NS, U.K.
T.Bending@mdx.ac.uk d.g.flaass@writeme.com

[Submitted: March 25, 1998; Accepted: June 30, 1998.]

[1991 Mathematical Subject Classification: 05E30, 05B20]

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() + Q() + Q(x + + ) 6= 0 for any three distinct x; ; ; Q(x) + Q() + Q() + Q(x + a) + Q( + a) + Q( + a) 6= 0 if a 6= 0 (x; ; arbitrary). We show that every crooked function gives rise to a distance regular graph of diameter 3 having = 0 and = 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