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.