In this paper, cyclic codes are studied over some finite quaternion integer rings with respect to the quaternion Mannheim distance.

In this study, we obtain new classes of linear codes over Hurwitz integers equipped with a new metric. We refer to the metric as Hurwitz metric. The codes with respect to Hurwitz metric use in coded modu- lation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither Hamming metric nor Lee metric. Also, we define decoding algorithms for these codes when up to two coordinates of a transmitted code vector are effected by error of arbitrary Hurwitz weight.Comment: 11 page

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In this paper, some nonbinary quantum codes using classical codes over Gaussian integers are obtained. Also, some of our quantum codes are better than or comparable with those known before, (for instance $[[8,2,5]]_{4+i}$).

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