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小字母表上完全互补码的构造

Construction of Complete Complementary Codes over Small Alphabet

Palash Sarkar, Chunlei Li, Sudhan Majhi, Zilong Liu

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AI总结：

本文研究小字母表上完全互补码的构造问题，通过在特定域上定义q元函数，提出了具有灵活长度、集合大小和字母表大小的传统CCCs及频谱零约束CCCs的构造方法。

摘要：

完全互补码（CCCs）不仅在无线通信中发挥着至关重要的作用，特别是在实现无干扰环境至关重要的多载波系统中，而且在构造其他需要适当函数以满足当今无线通信评估领域中各种需求的码方面也起着重要作用。本研究专注于为具有灵活长度、集合大小和字母表的传统和频谱零约束（SNC）CCCs构造$q$元函数。我们构造了传统CCCs，其长度定义为$L = \prod_{i=1}^k p_i^{m_i}$，集合大小定义为$K = \prod_{i=1}^k p_i^{n_i+1}$，字母表大小为$q=\prod_{i=1}^k p_i$，使得$p_1<p_2<\cdots<p_k$。参数$m_1, m_2, \ldots, m_k$（每个大于或等于$2$）是正整数，而$n_1, n_2, \ldots, n_k$是满足$n_i \leq m_i-1$的非负整数，变量$k$表示正整数。为了实现这些特定参数，我们在域$\mathbf{Z}_{p_1}^{m_1}\times \cdots \times \mathbf{Z}_{p_k}^{m_k}$上定义$q$元函数，该域被认为是$\mathbb{Z}_{q}^m$的适当子集，并包含$\prod_{i=1}^k p_i^{m_i}$个向量，其中$\mathbf{Z}_{p_i}^{m_i}=\{0,1,\hdots,p_i-1\}^{m_i}$，$m$的值由$m_1, m_2, \ldots, m_k$之和得出。这种域的组织使我们能够涵盖字母表$\mathbb{Z}_q$上所有可能的整数值长度序列。已经证明，通过约束生成传统CCCs的$q$元函数，我们可以推导出具有相同长度和字母表但集合大小小于或等于传统CCCs的SNC-CCCs。

英文摘要：

Complete complementary codes (CCCs) play a vital role not only in wireless communication, particularly in multicarrier systems where achieving an interference-free environment is of paramount importance, but also in the construction of other codes that necessitate appropriate functions to meet the diverse demands within today's landscape of wireless communication evaluation. This research is focused on the area of constructing $q$-ary functions for both of {traditional and spectrally null constraint (SNC) CCCs}\footnote{When no codes in CCCs having zero components, we call it as traditonal CCCs, else, we call it as SNC-CCCs in this pape.} of flexible length, set size and alphabet. We construct traditional CCCs with lengths, defined as $L = \prod_{i=1}^k p_i^{m_i}$, set sizes, defined as $K = \prod_{i=1}^k p_i^{n_i+1}$, and an alphabet size of $q=\prod_{i=1}^k p_i$, such that $p_1<p_2<\cdots<p_k $. The parameters $m_1, m_2, \ldots, m_k$ (each greater than or equal to $2$) are positive integers, while $n_1, n_2, \ldots, n_k$ are non-negative integers satisfying $n_i \leq m_i-1$, and the variable $k$ represents a positive integer. To achieve these specific parameters, we define $q$-ary functions over a 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 $m$ is derived from the sum of $m_1, m_2, \ldots, m_k$. This organization of the domain allows us to encompass all conceivable integer-valued length sequences over the alphabet $\mathbb{Z}_q$. It has been demonstrated that by constraining a $q$-ary function that generates traditional CCCs, we can derive SNC-CCCs with identical length and alphabet, yet a smaller or equal set size compared to the traditional CCCs.

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