Ovals in the Designs W$(2^m)$

Abstract

Using the permutation action of the group ${PSL}_2(2^m)$ on its dihedral subgroups of order $2(2^m+1)$ for the description of the class of designs $W(2^m)$ derived from regular ovals in the Desarguesian projective plane of order $2^m$, we construct a $2$-design of ovals for $W(2^m)$ for $m\ge 3$, and thus determine certain properties of the binary codes of these classes of designs.