Characterizing the Hermitian and Ree unitals on 28 points (Q1378864)

A unital is a 2-\((q^3+1,q+1,1)\) design. The authors study the case \(q=3\), where many examples of unitals are known, and show that the dimension of the corresponding binary code is at least \(19\), with equality if and only if the design is the Ree unital. This confirms a conjecture of \textit{A. E. Brouwer} [Geometries and groups, Proc. Colloq., Berlin 1981, Lect. Notes Math. 893, 183-188 (1981; Zbl 0557.51002)]. For unitals without ovals, the number \(19\) may be replaced by \(21\), and equality occurs if and only if the design is the classical Hermitian unital.