Structure and Rank of a Cyclic Code over a Class of Nonchain Rings

Abstract

The rings $Z_4+\nu Z_4$ have been classified into chain rings and nonchain rings based on the values of ${\nu }^2\in Z_4+\nu Z_4$. In this paper, the structure of a cyclic code of arbitrary length over the rings $Z_4+\nu Z_4$ for those values of ${\nu }^2$ for which these are nonchain rings has been established. A unique form of generators for a cyclic code over these rings has also been obtained. Furthermore, the rank and cardinality of a cyclic code over these rings have been established by finding a minimal spanning set for the code.