New distance regular graphs arising from dimensional dual hyperovals

In [4] we have studied the semibiplanes Sigma (e)(m,h) = Af(S-m,h(e)) 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 by re the incidence graph of (the point-block system of) Sigma (e)(m,h), we prove that Gamma (e)(m,h) is distance regular if and only if either m + h = e or (m + h, e) = 1. In the latter case, Gamma (e)(m,h) has the same array as the coset graph K-h(e) of the extended binary Kasami code K(2(e), 2(h)) but, as we prove in this paper, we have Gamma (e)(m,h) similar or equal to K-h(e) if and only if m = h. Finally, by exploiting some information obtained on Gamma (e)(m,h), we prove that if e less than or equal to 13 and m not equal h with (m + h, e) = 1, then Sigma (e)(m,h) is simply connected. (C) 2001 Academic Press.