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.
Acknowledgments
Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality.
Disclosure statement
No potential conflict of interest was reported by the authors.