Further results on association schemes from non-weakly regular bent functions

Abstract Recent studies have shown that the preimage set partitions of weakly regular bent functions, particularly those that are vectorial dual–bent, can give rise to association schemes. The first construction of association schemes from non-weakly regular bent functions, namely from ternary generalized Maiorana–McFarland bent functions from F₃䂞 F₃㵮 F₃㵮 to F₃ for n = 1 and n = 2, is presented in Özbudak and Pelen (J. Algebraic Combin. 56 (2022), 635–658). This construction was substantially generalized to arbitrary odd primes p and positive integers n in a recent work of Anbar et al. (Finite Fields Appl. 103 (2025), Paper No. 102568), using a variant of generalized Maiorana–McFarland bent functions. In this paper, we refine the construction of Anbar et al. to obtain association schemes on F₏^₍ F₏^₊ F₏^₊ with (p^k - 1p - 1 (p + 1) + p), (p^k - 1p - 1 (p + 1) + p - 1) \; and (p^k - 1p - 1 (p + 1) + p - 12) association classes, depending on n and on the choice of bent functions employed in the construction. We further emphasize that the association schemes previously obtained in the works of Özbudak–Pelen and Anbar et al. arise as fusion schemes of those constructed in this paper.