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 ${\mathbf{2}}^{(m-1)}\times 2$ 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 ${\mathbf{2}}^{(m-1)}\times 2$ respectively.