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Universal Costas Matrices: Towards a General Framework for Costas Array Construction

Fatih Gulec, Vahid Abolghasemi

TL;DR

This work introduces Universal Costas Matrices (UCMs) and Universal Costas Frequency Matrices (UCFMs) as a unified, matrix-based framework to analyze and discover Costas arrays. It formalizes the structure of UCMs and UCFMs, proves key symmetry properties, and proposes a reconstruction algorithm to derive Costas arrays from their frequency representations. By leveraging complete and incomplete UCFMs as training data, the paper lays the groundwork for AI-assisted Costas array discovery, achieving significant runtime improvements over traditional exhaustive-search baselines. The framework supports scalable generation of Costas arrays with potential impact on ISAC applications by delivering arrays with ideal autocorrelation and low cross-correlation. Overall, the work bridging combinatorial Costas array theory with data-driven discovery offers a path toward generalized construction beyond traditional algebraic methods.

Abstract

Costas arrays are a special type of permutation matrices with ideal autocorrelation and low cross-correlation properties, making them valuable for radar, wireless communication, and integrated sensing and communication applications. This paper presents a novel unified framework for analyzing and discovering new Costas arrays. We introduce Universal Costas Matrices (UCMs) and Universal Costas Frequency Matrices (UCFMs) and investigate their structural characteristics. A framework integrating UCMs and UCFMs is proposed to pave the way for future artificial intelligence-assisted Costas array discovery. Leveraging the structural properties of UCMs and UCFMs, a reconstruction-based search method is developed to generate UCMs from UCFMs. Numerical results demonstrate that the proposed approach significantly accelerates the search process and enhances structural insight into Costas array generation.

Universal Costas Matrices: Towards a General Framework for Costas Array Construction

TL;DR

This work introduces Universal Costas Matrices (UCMs) and Universal Costas Frequency Matrices (UCFMs) as a unified, matrix-based framework to analyze and discover Costas arrays. It formalizes the structure of UCMs and UCFMs, proves key symmetry properties, and proposes a reconstruction algorithm to derive Costas arrays from their frequency representations. By leveraging complete and incomplete UCFMs as training data, the paper lays the groundwork for AI-assisted Costas array discovery, achieving significant runtime improvements over traditional exhaustive-search baselines. The framework supports scalable generation of Costas arrays with potential impact on ISAC applications by delivering arrays with ideal autocorrelation and low cross-correlation. Overall, the work bridging combinatorial Costas array theory with data-driven discovery offers a path toward generalized construction beyond traditional algebraic methods.

Abstract

Costas arrays are a special type of permutation matrices with ideal autocorrelation and low cross-correlation properties, making them valuable for radar, wireless communication, and integrated sensing and communication applications. This paper presents a novel unified framework for analyzing and discovering new Costas arrays. We introduce Universal Costas Matrices (UCMs) and Universal Costas Frequency Matrices (UCFMs) and investigate their structural characteristics. A framework integrating UCMs and UCFMs is proposed to pave the way for future artificial intelligence-assisted Costas array discovery. Leveraging the structural properties of UCMs and UCFMs, a reconstruction-based search method is developed to generate UCMs from UCFMs. Numerical results demonstrate that the proposed approach significantly accelerates the search process and enhances structural insight into Costas array generation.
Paper Structure (10 sections, 5 theorems, 19 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 5 theorems, 19 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Universal Costas Matrices
  3. Preliminaries
  4. Structural Properties and Theoretical Results
  5. Universal Costas Frequency Matrices
  6. A General Framework for Costas Array Discovery: Reconstruction of UCMs from UCFMs
  7. Motivation
  8. Reconstruction Algorithm
  9. Results
  10. Discussion and Conclusion

Key Result

Theorem 1

In a universal Costas matrix $U_n$, all columns have equal sums, i.e.,

Figures (3)

  • Figure 1: A general framework to discover new Costas arrays using the UCMs/UCFMs and AI.
  • Figure 2: Benchmark comparison results of the proposed UCM reconstruction method with the Russo method.
  • Figure 3: Complete UCFMs ($F_n$) for $5 \leq n \leq 29$ (rows 1-5) and incomplete UCFMs for $n={30, 45, 176, 230, 248}$ (only row 6).

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Definition 4
  • Example 2
  • ...and 6 more