Almost Two-Valued Periodic Golay Sequence Pairs Derived From the Legendre Symbol

We describe a new family of unimodular almost two-valued periodic Golay sequence pairs of length p, where p is an odd prime. The sequences are obtained from the Legendre symbol by replacing the letters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0, 1, -1$ </tex-math></inline-formula> with the letters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1, A, B\in \Bbb {C}$ </tex-math></inline-formula>, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$|A|=|B|=1$ </tex-math></inline-formula>, where B additionally satisfies an appropriate constraint. The family is infinite in that each value of A yields at least one Golay pair or an ideal sequence. In special cases the family includes: 1) sequences with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$A=1$ </tex-math></inline-formula>, or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B=\pm \bar {A}$ </tex-math></inline-formula>, or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B=\pm i\bar {A}$ </tex-math></inline-formula>, or where both A and B are Gaussian rationals, or roots of unity; and 2), sequences whose autocorrelation sidelobes approach zero as p approaches infinity. These results extend the results of Björck on two-valued/almost two-valued ideal sequences derived from cyclic p-roots and the results of Golomb on two-valued ideal sequences derived from Hadamard-Paley difference sets. Related, but in some ways more constrained results on two-valued Golay sequence pairs derived from difference set families were recently published by Li et al.