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 \(m \geq 3\), and thus determine certain properties of the binary codes of these classes of designs.