On Buekenhout-Metz unitals of even order

Designs arising from Buekenhout‐Metz unitals

“…Then, to every (nondegenerate) affine Hermitian cone with vertex Y 1 , there is a uniquely associated pair ðw; vÞ with w 2 W and v 2 " C 0 , such that the affine points Pðx; yÞ of the Hermitian cone H w;v are those with ðx; yÞ satisfying the equation w q X q À wXþ v ¼ 0. Results on collineations which play an important role in the present investigation are now collected from previous papers [1,4]. Let G denote the group consisting of all linear collineations:…”

A characterization of the classical unital

“…Further characterizations of Buekenhout-Metz unitals in terms of their Baer sublines are given in [11], [5] and [15]. A complete characterization of Buekenhout-Metz unitals is given in [2] and [6]. Lef6vre-Percsy [13] and Faina and Korchm(tros [7] independently proved the following characterization of classical unitals.…”

Construction of unitals in the Hall planes

“…Meanwhile, ifq = 2 n _> 4, let tr: GF(q) ~ GF(2) with tr(x) = x + x 2 +... + x 2'~-1, C1 = {x C GF(q)ltr(x)= 1} andC0 = {x E GF(q)ltr(x)= 0} then a BM-unital is represented by Ha,b with b ~ GF(q) and aq+l(b q + b) -2 E Co (see [1], [4], [5]). For short, we will call these unitals the Buekenhout unitals.…”