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

Official URL: https://dx.doi.org/10.1016/j.ffa.2024.102568

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 Fpn×Fpk×Fpk to Fp, which are constructed from pk bent functions from Fpn to Fp 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 Fp to Fp respectively from Fp2 to Fp, 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.