## Abstract

In this paper, we study quasi-cyclic codes of index \(1\frac {1}{2}\) and co-index 2*m* over \(\mathbb {F}_{q}\) and their dual codes, where *m* is a positive integer, *q* is a power of an odd prime and \(\gcd (m,q) = 1\). We characterize and determine the algebraic structure and the minimal generating set of quasi-cyclic codes of index \(1\frac {1}{2}\) and co-index 2*m* over \(\mathbb {F}_{q}\). We note that some optimal and good linear codes over \(\mathbb {F}_{q}\) can be obtained from this class of codes. Furthermore, the algebraic structure of their dual codes is given.

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## Acknowledgments

The authors would like to thank the two anonymous reviewers and the Associate Editor for their valuable suggestions and comments that helped to greatly improve the paper. This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant No. 61571243), and the Fundamental Research Funds for the Central Universities of China.

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Gao, Y., Fang, W. & Fu, F. On the algebraic structure of quasi-cyclic codes of index \(1\frac {1}{2}\).
*Cryptogr. Commun.* **12, **1–18 (2020). https://doi.org/10.1007/s12095-019-0352-7

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### Keywords

- Quasi-cyclic codes of index \(1\frac {1}{2}\)
- Algebraic structure
- Minimal generating set
- Dual codes

### Mathematics Subject Classification (2010)

- 94B05
- 94B15
- 11T71