Cyclic codes over a non-commutative ring

Abstract

Quaternion ring with coefficient from ℤ 3 is a non-commutative finite ring. The structure of linear and cyclic codes over H 3 = ℤ 3 + ℤ 3 i + ℤ 3 j + ℤ 3 k is given. Also, a generator matrix in standard form for linear codes over the ring is given. It is shown that H 3 decomposes into two parts form ℤ 3 +ℤ 3 i 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.