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

TL;DR

This paper introduces a novel construction method for type-II Z-complementary code sets that can generate arbitrarily large sets with more codes than traditional type-I sets, using extended Boolean functions and graph theory.

Contribution

It presents a new construction for type-II ZCCS that surpasses the size limits of type-I ZCCS, enabling larger code sets with potential applications in communications.

Findings

Constructed type-II ZCCS with larger code sets than type-I.

Uses extended Boolean functions and Hamiltonian path properties.

Can generate $(p^k,p^k,p^n)$-CCC as a special case.

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 $k\le n$.