Unital

A $2$-$(q^3+1,q+1,1)$-design (cf. also Block design). It arose originally as the set of self-conjugate points and non-self-conjugate lines in a unitary polarity in a Desarguesian projective plane $\mathrm{PG}(2,q^2)$ (cf. Desargues geometry), in which case it has an automorphism group $\mathrm{P}\Gamma\mathrm{U}(3,q^2)$ with associated simple group $\mathrm{PSU}(3,q^2)$ (when $q>2$); see [a3]. This type is known as a classical or Hermitian unital. The points can be considered as points of the curve with equation $$ X^{q+1} + Y^{q+1} + Z^{q+1} = 0 $$ whose coordinates lie in the field $\mathrm{GF}(q^2)$. The design was first explicitly constructed by R.C. Bose [a2]. In fact, such a polarity and hence a unital exists in a non-Desarguesian plane constructed from a finite commutative semi-field with an involution [a6].

A unital with $q=6$ has been constructed by R. Mathon [a5] and by S. Bagchi and B. Bagchi [a1]. This shows that a unital of order $q$ cannot necessarily be embedded in a plane of order $q$. It had in fact been shown by H. Lüneberg [a4] that another class of unitals, the Ree unitals, having an associated simple automorphism group $G$, cannot be embedded in a projective plane in such a way that $G$ is induced by a collineation group of the plane.

A class of unitals other than the Hermitian ones and those embeddable in $\mathrm{PG}(2,q^2)$ are the Buekenhout–Metz unitals.

References

[a1]	S. Bagchi, B. Bagchi, "Designs from pairs of finite fields I. A cyclic unital $U(6)$ and other regular Steiner $2$-designs" J. Combin. Th. A , 52 (1989) pp. 51–61
[a2]	R.C. Bose, "On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements" , Golden Jubilee Commemoration Volume, 1958-1959 , Calcutta Math. Soc. (1959) pp. 341–354
[a3]	J.W.P. Hirschfeld, "Projective geometries over finite fields" , Oxford Univ. Press (1979)
[a4]	H. Lüneburg, "Some remarks concerning Ree groups of type $(G_2)$" J. Algebra , 3 (1966) pp. 256–259
[a5]	R. Mathon, "Constructions of cyclic 2-designs" Ann. Discrete Math. , 34 (1987) pp. 353–362
[a6]	F. Piper, "Unitary block designs" R.J. Wilson (ed.) , Graph Theory and Combinatorics , Research Notes in Mathematics , 34 , Pitman (1979) pp. 98–105