2024/09/22 by Palash Sarkar, Sarkar, Palash, Chunlei Li +5
Computer Science · Engineering · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2409.14462
openalex publication_date 2024/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This research focuses on constructing q-ary functions for complete complementary codes (CCCs) with flexible parameters. Most existing work has primarily identified sufficient conditions for q-ary functions related to q-ary CCCs. To the best of the authors' knowledge, this study is the first to establish both the necessary and sufficient conditions for q-ary functions, encompassing most existing CCCs constructions as special cases. For q-ary CCCs with a length of qm and a set size of qn+1, we begin by analyzing the necessary and sufficient conditions for q-ary functions defined over the domain ℤqm. Additionally, we construct CCCs with lengths given by L = ∏i=1k pimi, set sizes given by K = ∏i=1k pini+1, and an alphabet size of ν= ∏i=1k pi, where p1 < p2 < ⋯ < pk. To achieve these specific parameters, we examine the necessary and sufficient conditions for ν-ary functions over the domain Zp1m1 × ⋯ × Zpkmk, which is a subset of ℤνm and contains ∏i=1k pimi vectors. In this context, Zpimi = \0, 1, …, pi - 1\mi, and m is the sum of m1, m2, …, mk. The q-ary and ν-ary functions allow us to cover all possible length sequences. However, we find that the proposed ν-ary functions are more suitable for generating CCCs with a length of L = ∏i=1k pimi, particularly when mi is coprime to mj for some 1 ≤ i ≠ j ≤ k. While the proposed q-ary functions can also produce CCCs of the same length L, the set size and alphabet size become as large as L, since in this case, the only choice for q is L. In contrast, the proposed ν-ary functions yield CCCs with a more flexible set size K≤ L and an alphabet size of ν