Multivariable Function for New Complete Complementary Codes With Arbitrary Lengths

This paper is {devoted} to efficient design of complete complementary codes (CCCs) which have found wide applications in coding, signal processing and wireless communication, thanks to their {ideal} auto- and cross-correlation sum properties. A major motivation of this research is that the existing state-of-the-art can generate CCCs with certain lengths only and therefore may not {be able to meet} the diverse requirements in practice. We introduce a new tool called multivariable functions {with which we} propose a direct construction of CCCs with any \textit{arbitrary} lengths in the form of $\prod_{i=1}^k p_i^{m_i}$, where $k$ is a positive integer, $p_1,p_2,\hdots,p_k$ are prime numbers, and $m_1,m_2,\hdots,m_k$ are positive integers. The proposed set size is given by $\prod_{i=1}^k p_i^{n_i}$ ($n_1,n_2,\hdots,n_k$ positive integers), which covers all possible positive integers greater than 1. For $k=1$ and $p_1=2$, our proposed {construction reduces} to the exact Golay-Davis-Jedwab (GDJ) sequence generator as a special case. For $k>1$ and $p_1=p_2=\cdots=p_k=2$, it gives rise to the conventional CCCs with power-of-two lengths obtained from generalized Boolean functions. {Moreover, we introduce a linear code in connection with the proposed sets of CCCs.}