Boolean functions of binary Type-II and Type-III/II complementary array pairs

The sequence pairs of length \(2^{m}\) $2^m$ projected from Type-II and Type-III/II complementary array pairs of size \(2\times 2\times \cdots \times 2\) $2\times 2\times \cdots \times 2$ (m-times) form Type-II and Type-III complementary sequence pairs, respectively. An exhaustive search for binary Type-II and Type-III complementary sequence pairs of small lengths \(2^{m}\) $2^m$ ( \(m=1,2,3,4\) $m=1,2,3,4$ ) 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 Type-II and Type-III/II complementary array pairs of size \(2\times 2\times \cdots \times 2\) $2\times 2\times \cdots \times 2$ .