sentence1: in , it was proven that some of the best non-linear codes , such as the kerdock , preparata , and goethal codes can be viewed as linear codes over z 4 via the gray map from z n 4 to f 2n 2 .
sentence2: in , hammons et al showed that some important binary nonlinear codes can be obtained from cyclic codes over z 4 through the gray map .