Legendre pairs of lengths $\ell \equiv 0$ (mod $3$)

TL;DR

This paper establishes a connection between constant-PAF sequences and Legendre pairs with integer PSD values, introduces algorithms for spectral analysis, and proves the existence of Legendre pairs for specific lengths, advancing the understanding of their existence.

Contribution

It introduces two new number-theoretic algorithms to analyze Legendre pairs and proves their existence for several previously unresolved lengths.

Findings

Legendre pairs of lengths 117, 129, 133, and 147 exist.

Complete spectral characterization for lengths divisible by 3.

Identifies 12 lengths under 200 with unresolved Legendre pair existence.

Abstract

We prove a proposition that connects constant-PAF sequences and the corresponding Legendre pairs with integer PSD values. We show how to determine explicitly the complete spectrum of the $(\ell/3)$-rd value of the discrete Fourier transform for Legendre pairs of lengths $\ell \equiv 0 \, (\mod 3)$. This is accomplished by two new algorithms based on number-theoretic arguments. As an application, we prove that Legendre pairs of the open lengths 117, 129, 133, and 147 exist by finding Legendre pairs of these lengths with a multiplier group of order at least 3. As a consequence, 85, 87, 115, 145, 159, 161, 169, 175, 177, 185, 187, 195 are the twelve integers in the range < 200 for which the question of existence of Legendre pairs remains unsolved.