Abstract
Let q be an odd prime power, and denote by \({\mathbb {F}}_q\) the finite field with q elements. In this paper, we consider the ring \(R={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q\), where \(u^2=u, v^2=v,uv=vu=0\) and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over \({\mathbb {F}}_5\) as the Gray images of these codes over R are given in short lengths.
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Acknowledgements
The authors are thankful to the University Grants Commission (UGC), Govt. of India for financial support and Indian Institute of Technology Patna for research facilities. Also, authors would like to thank the anonymous referee(s) and the Editor for their valuable comments to improve the presentation of the manuscript.
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Yadav, S., Islam, H., Prakash, O. et al. Self-dual and LCD double circulant and double negacirculant codes over \({\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q\). J. Appl. Math. Comput. 67, 689–705 (2021). https://doi.org/10.1007/s12190-021-01499-9
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DOI: https://doi.org/10.1007/s12190-021-01499-9