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

TL;DR

This paper proves that all binary complementary array pairs of certain sizes are characterized by specific algebraic normal forms, extending previous exhaustive search results to a general proof.

Contribution

The paper provides a general proof that the algebraic normal forms characterize all binary Type-II and mixed Type-II/III complementary array pairs of the specified sizes.

Findings

All binary Type-II complementary array pairs are characterized by specific algebraic normal forms.

All mixed Type-II/III complementary array pairs of the given size are also characterized by these forms.

The proof extends previous exhaustive search results to a general theoretical framework.

Abstract

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.