Sie können Bookmarks mittels Listen verwalten, loggen Sie sich dafür bitte in Ihr SLUB Benutzerkonto ein.
Medientyp:
E-Artikel
Titel:
Legendre pairs of lengths ℓ ≡ 0 (mod 5)
Beteiligte:
Kotsireas, Ilias S.;
Koutschan, Christoph;
Bulutoglu, Dursun A.;
Arquette, David M.;
Turner, Jonathan S.;
Ryan, Kenneth J.
Erschienen:
Walter de Gruyter GmbH, 2023
Erschienen in:Special Matrices
Sprache:
Englisch
DOI:
10.1515/spma-2023-0105
ISSN:
2300-7451
Entstehung:
Anmerkungen:
Beschreibung:
<jats:title>Abstract</jats:title>
<jats:p>By assuming a type of balance for length <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0105_eq_003.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi>ℓ</m:mi>
<m:mo>=</m:mo>
<m:mn>87</m:mn>
</m:math>
<jats:tex-math>\ell =87</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0105_eq_004.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi>ℓ</m:mi>
<m:mo>=</m:mo>
<m:mn>85</m:mn>
</m:math>
<jats:tex-math>\ell =85</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, we find LPs of these lengths. We then study the power spectral density (PSD) values of <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0105_eq_005.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi>m</m:mi>
</m:math>
<jats:tex-math>m</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> compressions of LPs of length <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0105_eq_006.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mn>5</m:mn>
<m:mi>m</m:mi>
</m:math>
<jats:tex-math>5m</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. We also formulate a conjecture for LPs of lengths <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0105_eq_007.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi>ℓ</m:mi>
<m:mo>≡</m:mo>
<m:mn>0</m:mn>
</m:math>
<jats:tex-math>\ell \equiv 0</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> (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 <jats:inline-formula>
<jats:alternatives>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_spma-2023-0105_eq_008.png" />
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mo>≤</m:mo>
<m:mn>200</m:mn>
</m:math>
<jats:tex-math>\le 200</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> for which the existence question of LPs remains unsolved from 12 to 10.</jats:p>