Gray Images of Constacyclic Codes over a Non-chain Extension of Z 4

Let [Formula: see text] be the ring of integers modulo 4. We study the [Formula: see text]-constacyclic and [Formula: see text]-cyclic codes over the non-chain ring [Formula: see text] for a unit [Formula: see text] in [Formula: see text]. We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over [Formula: see text] are cyclic, quasi-cyclic or permutation equivalent to this code. For an odd positive integer [Formula: see text], we determine the generator polynomials of cyclic and [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text]. Further, we prove that a [Formula: see text]-cyclic code of length [Formula: see text] is a [Formula: see text]-constacyclic code if [Formula: see text] is odd, and a [Formula: see text]-quasi-twisted code if [Formula: see text] is even. A few examples are also incorporated, in which two parameters are new and one is best known to date.