On the Cyclic and-Constacyclic Codes over the Quaternion Rings and Their Applications

In this paper, the structures of linear codes over the quaternion rings with coe cient from Zp, Hp = Zp+Zpi+Zpj+Zpk are given, where p is an odd prime, i2 = j2 = k2 = p 1 and ij = (p 1)ji = k. The quaternion rings over Zp decompose into two parts form Zp + iZp(or Zp + jZp or Zp +kZp) with idempotent coe cients, depending on selecting a central orthogonal idempotent pair. The structures of cyclic and-constacyclic codes over Hp are determined, where p 3(mod 4), p is an odd prime, is a unit in Hp and some examples are given. The duals of linear codes over Hp are investigated. The parameters of quantum codes are obtained from cyclic codes and-constacyclic over Hp.