The Gray Image of Codes over Finite Chain Rings

The results of J. F. Qiann et al. [4] on $(1-\gamma )$-cyclic codes over finite chain rings of nilpotency index 2 are extended to $(1-{\gamma }^e)$-cyclic codes over finite chain rings of arbitrary nilpotency index $e+1$. The Gray map is introduced for this type of rings. We prove that the Gray image of a linear $(1-{\gamma }^e)$-cyclic code over a finite chain ring is a distance-invariant quasi-cyclic code over its residue field. When the length of codes and the characteristic of a ring are relatively prime, the Gray images of a linear cyclic code and a linear $(1+{\gamma }^e)$-cyclic code are permutatively to quasi-cyclic codes over its residue field.