New Distance Regular Graphs Arising from Dimensional Dual Hyperovals

In 4 we have studied the semibiplanes Σm,he = Af(Sm,he) obtained as affine expansions of the d -dimensional dual hyperovals of Yoshiara 6. We continue that investigation here, but from a graph theoretic point of view. Denoting byΓm, he the incidence graph of (the point-block system of)Σm, he, we prove that Γm,heis distance regular if and only if eitherm + h = e or (m + h,e ) = 1. In the latter case, Γm,hehas the same array as the coset graph Kheof the extended binary Kasami code K(2e, 2h) but, as we prove in this paper, we have Γm, he ∼ = Kheif and only if m = h. Finally, by exploiting some information obtained on Γm, he, we prove that if e ≤ 13 and m ≠ = h with (m + h, e) = 1, then Σm,heis simply connected.