Cyclic codes over a non-commutative ring

Quaternion ring with coefficient from ℤ3is a non-commutative finite ring. The structure of linear and cyclic codes over H3= ℤ3+ ℤ3i + ℤ3j + ℤ3k is given. Also, a generator matrix in standard form for linear codes over the ring is given. It is shown that H3decomposes into two parts form ℤ3+ℤ3i with idempotent coefficients. Notice that the parts are commutative. We give the necessary and sufficient condition of being a cyclic code over the ring. Further, we give a generator polynomial for a cyclic code and get parameters of it.