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