Legendre pairs of lengths ℓ≡0(mod3)
Ilias Kotsireas, Christoph Koutschan · Journal of Combinatorial Designs · 2021
Abstract We prove a proposition that connects constant‐periodic autocorrelation function sequences and the corresponding Legendre pairs with integer power spectral density values. We show how to determine explicitly the complete spectrum of the ‐rd value of the discrete Fourier transform for Legendre pairs of lengths . 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 12 integers in the range for which the question of existence of Legendre pairs remains unsolved.