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

Abstract

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 $0, 1, -1$ with the letters $1, A, B\in \Bbb {C}$ , $|A|=|B|=1$ , 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 $A=1$ , or $B=\pm \bar {A}$ , or $B=\pm i\bar {A}$ , 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.