Quaternary Codes and Biphase Codes 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 $\mathbb Z_8$-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 $\mathbb Z_4$-codes are not linear. As applications, we construct families of low-correlation quadriphase and biphase sequences.