Abstract
In this paper, we discuss double circulant and double negacirculant codes over a family of finite rings \(R_{t} = \mathbb {F}_{q} + v_{1} \mathbb {F}_{q} + v_{2}\mathbb {F}_{q} + {\cdots } + v_{t}\mathbb {F}_{q}\); \(({v_{i}^{2}} = v_{i}, v_{i}v_{j} = v_{j}v_{i}=0, i,j= 1,2,\ldots ,t, i \neq j)\), where q is an odd prime power. We obtain necessary and sufficient conditions for double circulant codes (double negacirculant codes) to be self-dual codes and to be linear codes with complementary dual (or LCD codes) codes and study the algebraic structure of self-dual and LCD double circulant (double negacirculant codes) codes. We derive a formula to determine the total number of self-dual and LCD double circulant codes (double negacirculant codes) over the ring Rt. We also find distance bounds for double circulant codes over Rt. Moreover, we use a Gray map to prove that the families of self-dual and LCD double circulant codes under this Gray map are asymptotically good.
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Abbreviations
- LCD:
-
Linear with complementary dual
- CRT:
-
Chinese Remainder Theorem
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Acknowledgements
The authors are grateful to the anonymous reviewers who have given us helpful comments to improve the manuscript. A part of this paper was written during a stay of H. Q. Dinh and B. P. Yadav in the Vietnam Institute For Advanced Study in Mathematics (VIASM) in Summer 2022, they would like to thank the members of VIASM for their hospitality. B. P. Yadav also wants to thank CSIR for the financial support, file number 09/1023(0018)/2016 EMR-I. T. Bag’s research was supported by The Fields Institute when he was a postdoc fellow at Carleton University. D. Panario is partially funded by the Natural Science and Engineering Research Council of Canada, reference number RPGIN-2018-05328. A. K. Upadhaya thanks SERB DST for their support under the MATRICS scheme with file number MTR/2020/000006.
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Dinh, H.Q., Yadav, B.P., Bag, T. et al. Self-dual and LCD double circulant and double negacirculant codes over a family of finite rings \( \mathbb {F}_{q}[v_{1}, v_{2},\dots ,v_{t}]\). Cryptogr. Commun. 15, 529–551 (2023). https://doi.org/10.1007/s12095-022-00616-0
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DOI: https://doi.org/10.1007/s12095-022-00616-0