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Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Crooked Functions, Bent Functions, and Distance Regular Graphs. | Crooked Functions Bent Functions and Distance Regular Graphs T.D. Bending D. Fon-Der-Flaass School of Mathematical Sciences Queen Mary and Westheld 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 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 À 0 and y 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 a 2 V we denote by Ha Q or simply Ha the set Ha Ha Q Q x Q x a I x 2 Vg. We shall denote the size of a hnite set X either by XI or by X whichever notation looks better in the context. 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 5 1998 R34 2 DEFINITION 1 A mapping Q V W is called crooked if it satisfies the following three properties 1.1 Q 0 0 1.2 Q x y z x y z 0 for any three distinct x y z 1.3 Q x y z X a y a z a 0 if a 0. If Q V W is a crooked function and A 2 GL V B 2 GL W any two automorphisms then the function Q0 BQA Q0 x B Q A x is also crooked. We shall call such functions Q and Q0 equivalent. Also for every a 2 V the function Q x Q a x a is also crooked. We say that Q and every function equivalent to Q is affine equivalent to Q. PROPOSITION 2 If Q is a crooked mapping then 2.1 Q is a bijection 2.2 Every
