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

Ilias Kotsireas, Christoph Koutschan, Dursun Bulutoglu, David Arquette, Jonathan Turner, Kenneth Ryan

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.

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

Abstract

By assuming a type of balance for length and non-trivial subgroups of multiplier groups of Legendre pairs (LPs) for length , 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 (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 for which the existence question of LPs remains unsolved from 12 to 10.

Paper Structure

This paper contains 7 sections, 22 theorems, 57 equations, 4 tables.

Table of Contents

  1. Introduction
  2. LPs of lengths $\ell \equiv 0 \,\, (\mathrm{mod}\; 5)$
  3. Finding LPs of lengths $\ell = 85$ and $\ell=87$
  4. LPs of length $\ell = 85$
  5. LPs of length $\ell = 87$
  6. Searching for LPs of length $\ell=115$
  7. Conclusion and future research

Key Result

Theorem 1

Let $g \in \mathbb{Z}_\ell \rtimes \mathbb{Z}_\ell^\times \text{ and } {\boldsymbol{q}} \in \mathbb{C}^\ell$. Then $G_{g{\boldsymbol{q}}}=gG_{{\boldsymbol{q}}}g^{-1}$ and $|G_{{\boldsymbol{q}}}|=|G_{g{\boldsymbol{q}}}|$.

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • ...and 29 more