Classification of self dual quadratic bent functions

We classify all self dual and anti self dual quadratic bent functions in 2n variables under the action of the orthogonal group O(2n,\mathbb F₂){{O}(2n,\mathbb F_2)} . This is done through a classification of all 2n × 2n involutory alternating matrices over \mathbb F₂{\mathbb F_2} under the action of the orthogonal group. The sizes of the O(2n,\mathbb F₂){{O}(2n,\mathbb F_2)} -orbits of self dual and anti self dual quadratic bent functions are determined explicitly.