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A Direct Construction of 2D-CCC with Arbitrary Array Size and Flexible Set Size Using Multivariable Function

Gobinda Ghosh, Sachin Pathak

TL;DR

The paper addresses the need for flexible 2D-CCC designs without reliance on predetermined seeds. It introduces a multivariable-function (MVF) framework to directly construct 2D-CCC with arbitrary $m\times n$ array sizes and set sizes of the form $\prod_{i} p_i^{k_i}\prod_{j} q_j^{l_j}$, unifying and extending many prior 1D/2D complementary code families. A key contribution is Theorem 1, establishing a 2D $(\alpha,\alpha,m,n)$-CCC with the MVF-based construction, along with a PMEPR bound showing row PMEPR $\le \max\{q_j^{l_j}\}$ and column PMEPR $\le \max\{p_i^{k_i}\}$. The framework also generalizes existing GCAS/CCC constructions as special cases and has practical impact for OP-based massive MIMO URA systems due to reduced PMEPR and flexible design parameters.

Abstract

Recently, two-dimensional (2D) array codes have been found to have applications in wireless communication.In this paper, we propose direct construction of 2D complete complementary codes (2D-CCCs) with arbitrary array size and flexible set size using multivariable functions (MVF). The Peak-to-mean envelope power ratio (PMEPR) properties of row and column sequences of the constructed 2D-CCC arrays are investigated. The proposed construction generalizes many of the existing state-of-the-art such as Golay complementary pair (GCP), one-dimensional (1D)-CCC, 2D Golay complementary array set (2D-GCAS), and 2D-CCC with better parameters compared to the existing work.

A Direct Construction of 2D-CCC with Arbitrary Array Size and Flexible Set Size Using Multivariable Function

TL;DR

The paper addresses the need for flexible 2D-CCC designs without reliance on predetermined seeds. It introduces a multivariable-function (MVF) framework to directly construct 2D-CCC with arbitrary array sizes and set sizes of the form , unifying and extending many prior 1D/2D complementary code families. A key contribution is Theorem 1, establishing a 2D -CCC with the MVF-based construction, along with a PMEPR bound showing row PMEPR and column PMEPR . The framework also generalizes existing GCAS/CCC constructions as special cases and has practical impact for OP-based massive MIMO URA systems due to reduced PMEPR and flexible design parameters.

Abstract

Recently, two-dimensional (2D) array codes have been found to have applications in wireless communication.In this paper, we propose direct construction of 2D complete complementary codes (2D-CCCs) with arbitrary array size and flexible set size using multivariable functions (MVF). The Peak-to-mean envelope power ratio (PMEPR) properties of row and column sequences of the constructed 2D-CCC arrays are investigated. The proposed construction generalizes many of the existing state-of-the-art such as Golay complementary pair (GCP), one-dimensional (1D)-CCC, 2D Golay complementary array set (2D-GCAS), and 2D-CCC with better parameters compared to the existing work.
Paper Structure (12 sections, 2 theorems, 68 equations, 3 figures, 1 table)

This paper contains 12 sections, 2 theorems, 68 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. One-Dimensional Sequences
  4. Two-Dimensional Array
  5. Multivariable Function ghosh2023optimal2dzcacs
  6. Peak-to-Mean Envelope Power Ratio (PMEPR) pai2022two
  7. Proposed Construction of 2D-CCC
  8. Bound for PMEPR
  9. Generalization of the proposed work
  10. Comparison with Previous Works
  11. Conclusion
  12. Appendix: Proof of Theorem \ref{['KB']}

Key Result

Theorem 1

We define $G_{t}=\{a^{\theta}_{t}:\theta\in \Theta\}$ and $\psi(G_{t})=\{\psi\left(a^{\theta}_{t}\right):\theta\in \Theta\}$. Then the set $\{\psi(G_{t}):t\in T\}$ forms a 2D $(\alpha,\alpha,m,n)-CCC$, where, $\alpha=\prod_{i=1}^{a}p^{k_{i}}_{i}\prod_{j=1}^{b}q^{l_{j}}_{j}$, $m=\prod_{i=1}^{a}p_{i}^

Figures (3)

  • Figure 1: Auto-correlation result of any set of array from $\mathcal{G},$
  • Figure 2: Cross-correlation result of any two sets of array from $\mathcal{G}.$
  • Figure 3: Row and column sequence IAPR/PMEPR

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3: Code (ghosh2022direct)
  • Definition 4: ACCS of code (ghosh2022direct)
  • Definition 5: MOGCS(ghosh2022direct)
  • Definition 6: das2020two
  • Definition 7
  • Definition 8: das2020two
  • Theorem 1
  • proof
  • ...and 7 more