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

DOI10.1002/JCD.21806arXiv2101.03116OpenAlexW3202612442MaRDI QIDQ6357859

Christoph Koutschan, Ilias S. Kotsireas

Publication date: 8 January 2021

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 $ellequiv0,(mod3)$. 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.

Full work available at URL: https://doi.org/10.1002/jcd.21806

Mathematics Subject Classification ID

Combinatorics (05-XX)

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