New Infinite Family of Sequences Almost Over Q 8 With Ideal Autocorrelation

Abstract

In this paper, we provide the left and right Parseval relations over the quaternions. Moreover, we construct a new infinite family of sequences almost over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q_{8} $ </tex-math></inline-formula> with period <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> by a generalization of Legendre mapping, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p\equiv 1 \pmod {8}$ </tex-math></inline-formula> is prime and the sequences have ideal autocorrelation. The sequences are symmetric when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p\equiv 1 \pmod {16}$ </tex-math></inline-formula> , and antisymmetric when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p\equiv 9 \pmod {16} $ </tex-math></inline-formula> .