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Set Size Bound for Aperiodic Z-Complementary Sets

Cheng-Yu Pai, Yu-Che Tung, Zhen-Ming Huang, Chao-Yu Chen

TL;DR

The paper resolves a long-standing conjecture on the set-size upper bound for aperiodic Z-complementary sets by proving $M\le\left\lfloor\dfrac{NL}{Z}\right\rfloor$ using a Welch-bound framework. It also introduces an algebraic construction of ZCSs based on extended generalized Boolean functions (EGBFs) that achieve this bound with new parameter regimes, yielding optimal ZCSs such as $(6,4,6,4)$ and $(9,4,9,4)$. This work clarifies the limits of ZCS size, demonstrates the bound's tightness, and expands design possibilities for ZCS-based applications. The results advance sequence design for multi-carrier systems and other areas requiring low aperiodic cross-correlation within a specified zero-correlation zone.

Abstract

The widely and commonly adopted upper bound on the set size of aperiodic Z-complementary sets (ZCSs) in the literature has been a conjecture. In this letter, we provide detailed derivations for this conjectured bound. A ZCS is optimal when its set size reaches the upper bound. Furthermore, we propose a new construction of ZCSs based on extended generalized Boolean functions (EGBFs). The proposed method introduces optimal ZCSs with new parameters.

Set Size Bound for Aperiodic Z-Complementary Sets

TL;DR

The paper resolves a long-standing conjecture on the set-size upper bound for aperiodic Z-complementary sets by proving using a Welch-bound framework. It also introduces an algebraic construction of ZCSs based on extended generalized Boolean functions (EGBFs) that achieve this bound with new parameter regimes, yielding optimal ZCSs such as and . This work clarifies the limits of ZCS size, demonstrates the bound's tightness, and expands design possibilities for ZCS-based applications. The results advance sequence design for multi-carrier systems and other areas requiring low aperiodic cross-correlation within a specified zero-correlation zone.

Abstract

The widely and commonly adopted upper bound on the set size of aperiodic Z-complementary sets (ZCSs) in the literature has been a conjecture. In this letter, we provide detailed derivations for this conjectured bound. A ZCS is optimal when its set size reaches the upper bound. Furthermore, we propose a new construction of ZCSs based on extended generalized Boolean functions (EGBFs). The proposed method introduces optimal ZCSs with new parameters.

Paper Structure

This paper contains 7 sections, 4 theorems, 28 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Definitions
  3. Generalized Boolean function
  4. Extended Generalized Boolean function
  5. Derivation of the Upper Bound on the Set Size of ZCSs
  6. Proposed Construction of ZCSs
  7. Conclusion

Key Result

Lemma 1

Welch A matrix $\bm X$ of size $\overline L\times \overline M$ can be represented by where $\bm s_v=( s_{0, v}, s_{1, v},\ldots, s_{\overline L-1, v})$ and $\sum\limits_{l=0}^{\overline L-1} |s_{l,v}|^2=E$, for $v=0,1,\ldots,\overline M-1$. Let $\delta_{\text{max}}=\max\limits_{v \neq t} \left| \sum\limits_{l=0}^{\overline L-1} s_{l,v} s_{l,t}^*\right|$ be the maximum value of the i

Figures (1)

  • Figure 1: Comparison of known upper bounds on the set size of ZCSs with $N=4$ and $L=6$.

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Definition 2
  • Remark 1
  • Theorem 2
  • Remark 2
  • Example 1