New distance regular graphs arising from dimensional dual hyperovals (Q5942876)

The article of Pasini and Yoshiara deals with the semibiplanes \(\Sigma^e_{m,h} = Af(S^e_{m,h})\) obtained as affine expansions of the \(d\)-dimensional dual hyperoval of Yoshiara. NEWLINENEWLINENEWLINEThe authors obtain the following results: NEWLINENEWLINENEWLINETheorem 1. Let \(\Gamma^e_{m,h}\) be the incidence graph of the point-block-system of \(\Sigma^e_{m,h}\). Then \(\Gamma^e_{m,h}\) is distance regular if and only if \(m+h=e\) or \((m+h,e) = 1\). NEWLINENEWLINENEWLINETheorem 2. If \((m+h,e) = 1\), then \(\Gamma^e_{m,h}\) has the same array as the coset graph \(K^e_h\) of the extended binary Kasami code \(K(2^e,2^h)\). Furthermore \(\Gamma^e_{m,h} \cong K^e_h\), if and only if \(m = h\). NEWLINENEWLINENEWLINEThe paper finishes with the following conjecture: If \(m \neq h\) and \((m + h, e) = 1\), then \(\Sigma^e_{m,h}\) is simply connected. The conjecture is proven for the case \(e \leq 13\).