On left quaternion codes

Abstract

Let $F_q$ be a finite field of cardinality q where q is a power of an odd prime integer, and denote the generalized quaternion group ${\mathit{\text{Q}}}_n$ by the presentation: ${\mathit{\text{Q}}}_n=⟨x,y\mid x^n=1,y^2=x^{n/2},yx{y}^{-1}=x^{-1}⟩$ where n is even and satisfies $\gcd (n,q)=1$. Left ideals of the group algebra $F_q{\mathit{\text{Q}}}_n$ are called left quaternion codes over $F_q$ of length $2n$, and abbreviated as left ${\mathit{\text{Q}}}_n$-codes. In this paper, a system theory for left ${\mathit{\text{Q}}}_n$-codes is developed only using finite field theory and basic theory of cyclic codes and skew cyclic codes. First, we prove that any left ${\mathit{\text{Q}}}_n$-code is a direct sum of concatenated codes with the inner code ${\mathsf{A}}_i$ and the outer code $C_i$, where ${\mathsf{A}}_i$ is a minimal self-reciprocal cyclic code over $F_q$ of length n and $C_i$ is a skew constacyclic code of length 2 over an extension field or an extension principal ideal ring of $F_q$. Then we give explicit expressions for outer codes in the concatenated codes, and present the dual code for any left ${\mathit{\text{Q}}}_n$-code precisely. Moreover, all distinct left ${\mathit{\text{Q}}}_{12}$-codes over $F_5$ and all distinct left ${\mathit{\text{Q}}}_{20}$-codes over $F_3$ are presented, respectively.

Keywords:

• left quaternion code
• concatenated structure
• skew constacyclic code
• cyclic code
• dual code

2010 AMS Subject Classifications:

• 94B05
• 94B15
• 11T71

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.

ORCID

Y. Cao http://orcid.org/0000-0002-3089-0046

Additional information

Funding

This research is supported in part by the National Key Basic Research Program of China (Grant No. 2013CB834204) and the National Natural Science Foundation of China (Grant Nos. 11471255,61171082).