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New Correlation Bound and Construction of Quasi-Complementary Code Sets

Palash Sarkar, Chunlei Li, Sudhan Majhi, Zilong Liu

TL;DR

This work addresses scalability in MC-CDMA by studying QCSSs formed from CCCs and deriving a new aperiodic correlation bound on the maximum cross- and auto-correlation magnitude $\theta$, tighter than prior bounds. It introduces a graphical, $q$-ary function-based construction that yields small alphabets $q$ (divisible by a prime $p$) while producing asymptotically optimal performance with respect to the new bound as length $L$ grows. The methodological core combines a weighted quadratic bound on $\theta$ with a Hamiltonian-path graph framework to construct $(p^{n+1}(p-1),p^{n+1},p^m,p^m)$-QCSSs from $(p^{n+1},p^m)$-CCCs, validated by explicit examples. The results have practical implications for interference mitigation and capacity scaling in MC-CDMA, enabling larger user sets with robust aperiodic correlations and bounded inter-set cross-talk.

Abstract

Quasi-complementary sequence sets (QCSSs) have attracted sustained research interests for simultaneously supporting more active users in multi-carrier code-division multiple-access (MC-CDMA) systems compared to complete complementary codes (CCCs). In this paper, we investigate a novel class of QCSSs composed of multiple CCCs. We derive a new aperiodic correlation lower bound for this type of QCSSs, which is tighter than the existing bounds for QCSSs. We then present a systematic construction of such QCSSs with a small alphabet size and low maximum correlation magnitude, and also show that the constructed aperiodic QCSSs can meet the newly derived bound asymptotically.

New Correlation Bound and Construction of Quasi-Complementary Code Sets

TL;DR

This work addresses scalability in MC-CDMA by studying QCSSs formed from CCCs and deriving a new aperiodic correlation bound on the maximum cross- and auto-correlation magnitude , tighter than prior bounds. It introduces a graphical, -ary function-based construction that yields small alphabets (divisible by a prime ) while producing asymptotically optimal performance with respect to the new bound as length grows. The methodological core combines a weighted quadratic bound on with a Hamiltonian-path graph framework to construct -QCSSs from -CCCs, validated by explicit examples. The results have practical implications for interference mitigation and capacity scaling in MC-CDMA, enabling larger user sets with robust aperiodic correlations and bounded inter-set cross-talk.

Abstract

Quasi-complementary sequence sets (QCSSs) have attracted sustained research interests for simultaneously supporting more active users in multi-carrier code-division multiple-access (MC-CDMA) systems compared to complete complementary codes (CCCs). In this paper, we investigate a novel class of QCSSs composed of multiple CCCs. We derive a new aperiodic correlation lower bound for this type of QCSSs, which is tighter than the existing bounds for QCSSs. We then present a systematic construction of such QCSSs with a small alphabet size and low maximum correlation magnitude, and also show that the constructed aperiodic QCSSs can meet the newly derived bound asymptotically.
Paper Structure (14 sections, 9 theorems, 165 equations, 6 figures, 10 tables)

This paper contains 14 sections, 9 theorems, 165 equations, 6 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Existing Works on the Constructions and the Correlation Bounds of QCSSs
  3. Motivations and Contributions
  4. Preliminaries
  5. Aperiodic Auto- and Cross-Correlation
  6. Sequences associated with $q$-ary Functions
  7. Quadratic Functions and Graphs
  8. Tighter Lower Bounds on the Maximum Correlation Magnitude of QCSSs
  9. Construction of Asymptotically Optimal QCSSs Comprised of CCCs
  10. Conclusion
  11. Proof of Corollary 1
  12. Proof of Theorem 2
  13. Proof of Proposition 1
  14. Correlation Lower Bound with Respect to the Positive-Cycle-of-a-Sine-Wave Weight Vector when $t\in [L+1:2L-1]$

Key Result

Lemma 1

Let $\mathcal{C}$ be a $(K,M,L,\theta)$-QCSS. Then where

Figures (6)

  • Figure 1: Graph of the function $x_0x_2+2x_2x_1+2x_1^2+x_2+1$
  • Figure 2: Correlation plot for $\mathcal{C}_k$
  • Figure 3: Comparison between the optimality factors $\rho_1$ and $\rho_2$ with respect to $((p-1)p,p,p,p)$-QCSS for $13\leq p<15000$
  • Figure 4: Correlation plot between the codes $\psi(C_0^1)$ and $\psi(C_1^2)$
  • Figure 5: Correlation plot
  • ...and 1 more figures

Theorems & Definitions (17)

  • Lemma 1: crlbzl
  • Example 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Remark 1
  • Corollary 2
  • Remark 2
  • Remark 3
  • Theorem 2
  • ...and 7 more