New Infinite Family of Sequences Almost Over Q8 With Ideal Autocorrelation

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
$Q_{8} $
with period
$p$
by a generalization of Legendre mapping, where
$p\equiv 1 \pmod {8}$
is prime and the sequences have ideal autocorrelation. The sequences are symmetric when
$p\equiv 1 \pmod {16}$
, and antisymmetric when
$p\equiv 9 \pmod {16} $
.