Arcs and Ovals in the Hermitian and Ree Unitals

The hermitian unitals U ( q ) and the Ree unitals RU ( q ) are examined for the existence of ovals and arcs. It is shown that U ( q ) does not have ovals for q > 2 and that RU ( q ), like U ( q ), is embedded in a much larger design with block intersections of cardinality ⩽ 2. Arcs of size 3 q + 1 are constructed for the Ree unitals RU ( q ); they are ovals only in the case q = 3. In this case, U (3) and RU (3) are embedded in the same design and its automorphism group, the symplectic group Sp (6, 2), contains the automorphism groups of both the unitals; the coding-theoretic aspects are elucidated.

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