Abstract
Let be a finite field of cardinality q where q is a power of an odd prime integer, and denote the generalized quaternion group by the presentation: where n is even and satisfies . Left ideals of the group algebra are called left quaternion codes over of length , and abbreviated as left -codes. In this paper, a system theory for left -codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left -code is a direct sum of concatenated codes with the inner code and the outer code , where is a minimal self-reciprocal cyclic code over of length n and is a skew constacyclic code of length 2 over an extension field or an extension principal ideal ring of . Then we give explicit expressions for outer codes in the concatenated codes, and present the dual code for any left -code precisely. Moreover, all distinct left -codes over and all distinct left -codes over are presented, respectively.
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