Legendre pairs of lengths $\ell\equiv0$ (mod 5)

TL;DR

This paper introduces new Legendre pairs of lengths divisible by 5, develops a conjecture to streamline their search, and reduces the unresolved existence cases for lengths up to 200.

Contribution

It presents a novel approach for constructing Legendre pairs of lengths divisible by 5 and proposes a conjecture to optimize their search process.

Findings

Found new Legendre pairs of lengths 85 and 87.

Reduced unresolved cases for lengths ≤200 from 12 to 10.

Demonstrated the effectiveness of the conjecture in decreasing search space.

Abstract

By assuming a type of balance for length $\ell=87$ and non-trivial subgroups of multiplier groups of Legendre pairs (LPs) for length $\ell=85$, we find LPs of these lengths. We then study the power spectral density (PSD) values of m-compressions of LPs of length 5m. We also formulate a conjecture for Legendre pairs of lengths $\ell \equiv 0$ (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range $\leq 200$ for which the existence question of LPs remains unsolved from 12 to 10.