Some Good Cyclic and Quasi-Twisted ${\mathbb{Z}}_4$-Linear Codes

Abstract

For over a decade, there has been considerable research on codes over ${\mathbb{Z}}_4$ and other rings. In spite of this, no tables or databases exist for codes over ${\mathbb{Z}}_4$, as is the case with codes over finite fields. The purpose of this work is to contribute to the creation of such a database. We consider cyclic, negacyclic and quasi-twisted $(QT)$ codes over ${\mathbb{Z}}_4$. Some of these codes have binary images with better parameters than the best-known binary linear codes. We call such codes “good codes”. Among these are two codes which improve the bounds on the best-known binary non-linear codes. Tables of best cyclic and $QT$ codes over ${\mathbb{Z}}_4$ are presented.