We first define a new Gray map from to , where and study -constacyclic codes over R. Also of interest are some properties of -constacyclic codes over R. Considering their images, we prove that the Gray images of -constacyclic codes of length n over R are cyclic codes of length 2n over . In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic R-codes published by Özen et al. [7].