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New constructions of asymptotically optimal periodic and aperiodic quasi-complementary sequence sets

Peng Wang, Ziling Heng, Chengju Li

TL;DR

Compared with known asymptotically optimal aperiodic QCSSs in the literature, the proposed aperiodic QCSSs have better or new parameters.

Abstract

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code division multiple access (MC-CDMA) systems as they can support more users than perfect complementary sequence sets (PCSSs). The objective of this paper is to present new constructions of asymptotically optimal periodic and aperiodic QCSSs with large set sizes. Firstly, we construct a family of asymptotically optimal periodic $(p^{2n}, p^n-1, p^n-1, p^n+1)$ QCSSs with small alphabet size $p$, which has larger set size than the known family of periodic $(p^n(p^n-1), p^n-1, p^n-1, p^n+1)$ QCSSs. Secondly, we construct five new families of asymptotically optimal aperiodic QCSSs with large set sizes and low aperiodic tolerances. Each family of these aperiodic QCSSs has set size $Θ(K^2)$ for some flock size $K$. Compared with known asymptotically optimal aperiodic QCSSs in the literature, the proposed aperiodic QCSSs by us have better parameters or new lengths of their constituent sequences.

New constructions of asymptotically optimal periodic and aperiodic quasi-complementary sequence sets

TL;DR

Compared with known asymptotically optimal aperiodic QCSSs in the literature, the proposed aperiodic QCSSs have better or new parameters.

Abstract

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code division multiple access (MC-CDMA) systems as they can support more users than perfect complementary sequence sets (PCSSs). The objective of this paper is to present new constructions of asymptotically optimal periodic and aperiodic QCSSs with large set sizes. Firstly, we construct a family of asymptotically optimal periodic QCSSs with small alphabet size , which has larger set size than the known family of periodic QCSSs. Secondly, we construct five new families of asymptotically optimal aperiodic QCSSs with large set sizes and low aperiodic tolerances. Each family of these aperiodic QCSSs has set size for some flock size . Compared with known asymptotically optimal aperiodic QCSSs in the literature, the proposed aperiodic QCSSs by us have better parameters or new lengths of their constituent sequences.

Paper Structure

This paper contains 14 sections, 2 theorems, 93 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Periodic and aperiodic QCSSs
  4. The correlation lower bounds of periodic and aperiodic QCSSs
  5. Additive and multiplicative characters of finite fields
  6. The $m$-sequence
  7. A new construction of periodic QCSSs
  8. Five new constructions of aperiodic QCSSs
  9. The first construction of aperiodic QCSSs
  10. The second construction of aperiodic QCSSs
  11. The third construction of aperiodic QCSSs
  12. The fourth construction of aperiodic QCSSs
  13. The fifth construction of aperiodic QCSSs
  14. Concluding remarks

Key Result

Lemma 1

ZL1 For an aperiodic $(M, K ,N, \theta_{\max})$-QCSS with $M\geq3K$, $K\geq2$ and $N\geq2$, we have

Figures (12)

  • Figure 1: The auto-correlation magnitude distributions of $\mathbf{C}^{1,\alpha}$, $\mathbf{C}^{1,\alpha^2}$ and $\mathbf{C}^{\alpha,\alpha^2}$.
  • Figure 2: The cross-correlation magnitude distributions of $\mathbf{C}^{1,\alpha}$, $\mathbf{C}^{1,\alpha^2}$ and $\mathbf{C}^{\alpha,\alpha^2}$.
  • Figure 3: The auto-correlation magnitude distributions of $\mathbf{A}^{1,1}$, $\mathbf{A}^{1,2}$ and $\mathbf{A}^{2,1}$.
  • Figure 4: The cross-correlation magnitude distributions of $\mathbf{A}^{1,1}$, $\mathbf{A}^{1,2}$ and $\mathbf{A}^{2,1}$.
  • Figure 5: The auto-correlation magnitude distributions of $\mathbf{B}^{1,\alpha}$,$\mathbf{B}^{1,\alpha^2}$ and $\mathbf{B}^{2,\alpha}$.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6