Boolean Functions of Binary Type-II and Type-II/III Complementary Array Pair

The sequence pairs of length \(2^{m}\) projected from complementary array pairs of Type-II of size \(\mathbf{2}^{(m)}\) and mixed Type-II/III and of size \(\mathbf{2}^{(m-1)}\times2\) are complementary sequence pairs Type-II and Type-III respectively. An exhaustive search for binary Type-II and Type-III complementary sequence pairs of small lengths \(2^{m}\) (\(m=1,2,3,4\)) shows that they are all projected from the aforementioned complementary array pairs, whose algebraic normal forms satisfy specified expressions. It's natural to ask whether the conclusion holds for all \(m\). In this paper, we proved that these expressions of algebraic normal forms determine all the binary complementary array pairs of Type-II of size \(\mathbf{2}^{(m)}\) and mixed Type-II/III of size \(\mathbf{2}^{(m-1)}\times2\) respectively.