The $T_{4}$ and $G_{4}$ constructions of Costas arrays

TL;DR

This paper investigates two specific methods for constructing Costas arrays, linking them to Fibonacci primitive roots, and demonstrates their infinite validity under the Extended Riemann Hypothesis.

Contribution

It establishes the connection between the $T_{4}$ and $G_{4}$ constructions and Fibonacci primitive roots, proving their infinite occurrence under a major hypothesis.

Findings

The $T_{4}$ and $G_{4}$ constructions are valid infinitely often under the Extended Riemann Hypothesis.

Connection established between Costas array constructions and Fibonacci primitive roots.

Provides a theoretical foundation for the infinite existence of these constructions.

Abstract

We examine two particular constructions of Costas arrays known as the Taylor variant of the Lempel construction, or the $T_{4}$ construction, and the variant of the Golomb construction, or the $G_{4}$ construction. We connect these constructions with the concept of Fibonacci primitive roots, and show that under the Extended Riemann Hypothesis the $T_{4}$ and $G_{4}$ constructions are valid infinitely often.