Bent partitions and Maiorana-McFarland association schemes

The recently introduced generalized semifield spreads are partitions of FpmxFpm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{p<^>m}\times {\mathbb {F}}_{p<^>m}$$\end{document}, which are constructed from presemifields with a certain property, called right Fpk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{p<^>k}$$\end{document}-linearity. These partitions have similar properties as spreads. In particular, they are bent partitions, hence they yield a large number of bent functions, vectorial bent functions and amorphic association schemes. We show that with a slight change of parameters, we obtain inequivalent bent partitions, non-isomorphic divisible designs, and bent functions with various algebraic degrees. This is in contrast to classical spreads of FpmxFpm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{p<^>m} \times {\mathbb {F}}_{p<^>m}$$\end{document}, which yield bent functions all of which have algebraic degree (p-1)m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)m$$\end{document}. We show that with right Fpk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{p<^>k}$$\end{document}-linear presemifields we can obtain a large variety of vectorial dual-bent functions, which yield, not necessarily amorphic, association schemes. We investigate fusions of these association schemes, which reveal information on their inner structure, and may provide a tool to distinguish non-isomorphic association schemes.