Asymptotic bounds on the numbers of certain bent functions

Abstract

Using recent results of Keevash et al. [10] and Eberhard et al. [8] together with further new detailed techniques in combinatorics, we present constructions of two concrete families of generalized Maiorana-McFarland bent functions. Our constructions improve the lower bounds on the number of bent functions in n variables over a finite field \({\mathbb F}_p\) if p is odd and n is odd in the limit as n tends to infinity. Moreover we obtain the asymptotically exact number of two dimensional vectorial Maiorana-McFarland bent functions in n variables over \({\mathbb F}_2\) as n tends to infinity.