A Construction of Arbitrarily Large Type-II $Z$ Complementary Code Set

TL;DR

This paper addresses the need for larger code sets in zero-correlation-zone code sets by introducing a direct type-II $ZCCS$ construction using extended Boolean functions tied to a graph structure consisting of $k$ Hamiltonian paths and $r$ isolated vertices. The resulting family has parameters $\left(p^{k+r},\,p^k,\,p^{n+r}-p^r+1,\,p^{n+r}\right)$ with PMEPR bounded by $p$, and it includes $(p^k,p^k,p^n)$-CCC as a special case (when $r=0$). The construction subsumes several existing ZCCS/CChoosing frameworks (e.g., Shen, Rat, Davis) as special cases and demonstrates that type-II ZCCS can exceed type-I bounds in code count for the same $ZCZ$ width. By bridging EBFs, graph theory, and PMEPR considerations, the authors provide a flexible, scalable approach to interference-free MC-CDMA code design with practical impact on system capacity and robustness.

Abstract

For a type-I $(K,M,Z,N)$-ZCCS, it follows $K \leq M \left\lfloor \frac{N}{Z}\right\rfloor$. In this paper, we propose a construction of type-II $(p^{k+n},p^k,p^{n+r}-p^r+1,p^{n+r})$-$Z$ complementary code set (ZCCS) using an extended Boolean function, its properties of Hamiltonian paths and the concept of isolated vertices, where $p\ge 2$. However, the proposed type-II ZCCS provides $K = M(N-Z+1)$ codes, where as for type-I $(K,M,N,Z)$-ZCCS, it is $K \leq M \left\lfloor \frac{N}{Z}\right\rfloor$. Therefore, the proposed type-II ZCCS provides a larger number of codes compared to type-I ZCCS. Further, as a special case of the proposed construction, $(p^k,p^k,p^n)$-CCC can be generated, for any integral value of $p\ge2$ and $k\le n$.

Figures (1)

• Figure 1: Graph of two Hamiltonian paths and one isolated vertex.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof