Quaternary Codes and Biphase Sequences from $$\mathbb{Z}_8 $$ -Codes

Composing the Carlet map with the inverse Gray map, a new family of cyclic quaternary codes is constructed from 5-cyclic -codes. This new family of codes is inspired by the quaternary Shanbag–Kumar–Helleseth family, a recent improvement on the Delsarte–Goethals family. We conjecture that these -codes are not linear. As applications, we construct families of low-correlation quadriphase and biphase sequences.