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Legendre pairs of lengths $\ell \equiv 0$ (mod $3$)
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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 $\ell \equiv 0 \, (\mod 3)$. 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.
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Multiplier obstructions for Legendre pairs of length 333
A Legendre pair of length 333 cannot be invariant under any common multiplier subgroup of order at least 9; only subgroups of order at most 6 remain possible.
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