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Complete complementary codes (CCCs) play a vital role not only in wirelesscommunication, particularly in multicarrier systems where achieving aninterference-free environment is of paramount importance, but also in theconstruction of other codes that necessitate appropriate functions to meet thediverse demands within today's landscape of wireless communication evaluation.This research is focused on the area of constructing $q$-ary functions for bothof {traditional and spectrally null constraint (SNC) CCCs}\footnote{When nocodes in CCCs having zero components, we call it as traditonal CCCs, else, wecall it as SNC-CCCs in this pape.} of flexible length, set size and alphabet.We construct traditional CCCs with lengths, defined as $L = \prod_{i=1}^kp_i^{m_i}$, set sizes, defined as $K = \prod_{i=1}^k p_i^{n_i+1}$, and analphabet size of $q=\prod_{i=1}^k p_i$, such that $p_1<p_2<\cdots<p_k $. Theparameters $m_1, m_2, \ldots, m_k$ (each greater than or equal to $2$) arepositive integers, while $n_1, n_2, \ldots, n_k$ are non-negative integerssatisfying $n_i \leq m_i-1$, and the variable $k$ represents a positiveinteger. To achieve these specific parameters, we define $q$-ary functions overa domain $\mathbf{Z}_{p_1}^{m_1}\times \cdots \times \mathbf{Z}_{p_k}^{m_k}$that is considered a proper subset of $\mathbb{Z}_{q}^m$ and encompasses$\prod_{i=1}^k p_i^{m_i}$ vectors, where$\mathbf{Z}_{p_i}^{m_i}=\{0,1,\hdots,p_i-1\}^{m_i}$, and the value of $m$ isderived from the sum of $m_1, m_2, \ldots, m_k$. This organization of thedomain allows us to encompass all conceivable integer-valued length sequencesover the alphabet $\mathbb{Z}_q$. It has been demonstrated that by constraininga $q$-ary function that generates traditional CCCs, we can derive SNC-CCCs withidentical length and alphabet, yet a smaller or equal set size compared to thetraditional CCCs.

Construction of Complete Complementary Codes over Small Alphabet