The rings $Z_{4}+\nu Z_{4}$ have been classified into chain rings and non-chain rings on the basis of the values of $\nu^{2} \in Z_{4}+\nu Z_{4}.$ In this paper, the structure of cyclic codes of arbitrary length over the rings $Z_{4}+\nu Z_{4}$ for those values of $\nu^{2}$ for which these are non-chain rings has been established. A unique form of generators of these codes has also been obtained. Further, rank and cardinality of these codes have been established by finding minimal spanning sets for these codes.

翻译：基于$\nu^{2} \in Z_{4}+\nu Z_{4}$的取值，环$Z_{4}+\nu Z_{4}$被划分为链环和非链环两类。本文针对使$Z_{4}+\nu Z_{4}$成为非链环的$\nu^{2}$取值，建立了该环上任意长度循环码的结构，并得到了这些码生成元的一种唯一形式。进一步，通过寻找这些码的最小生成集，确定了它们的秩与基数。