(2p + 1)-class association schemes from the generalized Maiorana-McFarland class

Abstract

In several articles, it has been shown that the preimage set partition of weakly regular (vectorial) bent functions, which are vectorial dual-bent, give rise to association schemes. The first construction of association schemes from certain partitions obtained from non-weakly regular bent functions, namely from ternary generalized Maiorana-McFarland functions, is presented in Özbudak and Pelen (2022) [32].

In this article, association schemes are obtained from generalized Maiorana-McFarland bent functions from $F_{p^n}\times F_{p^k}\times F_{p^k}$ to $F_p$, which are constructed from $p^k$ bent functions from $F_{p^n}$ to $F_p$ with certain properties. The obtained schemes are in general $(2p+1)$-class association schemes. In the case that $n=1$ respectively in one case for $n=2$, the association schemes reduce to $(3p+1)/2$-class association schemes respectively to 2p-class association schemes. For $p=3$, these schemes are the 5-class and 6-class association schemes obtained by Özbudak and Pelen. Therefore, the construction in this article substantially generalizes these earlier constructions. Also note that for $n=1$ respectively $n=2$, the construction is based in bent functions from $F_p$ to $F_p$ respectively from $F_{p^2}$ to $F_p$, for which the choices are very limited.

Depending on the choice of the bent functions used for the construction, the resulting generalized Maiorana-McFarland function may be weakly regular or non-weakly regular, it may be an l-form, or it may not be an l-form. It is pointed out that for weakly regular bent l-forms, which can be obtained with this construction, the preimage set partition yields a p-class fusion scheme of a $(2p+1)$-class association scheme.