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On the number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\) and \({{\mathbb {Z}}}_{p}{{\mathbb {Z}}}_{p^{2}}\)-additive cyclic codes

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Abstract

In this paper, we give the exact number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\)-additive cyclic codes of length \(n=r+s,\) for any positive integer r and any positive odd integer s. We will provide a formula for the the number of separable \({{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n and then a formula for the number of non-separable \({{\mathbb {Z}} _{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n. Then, we have generalized our approach to give the exact number of \({{\mathbb {Z}}_{p}{\mathbb { Z}_{p^{2}}}}\)-additive cyclic codes of length \(n=r+s,\) for any prime p,  any positive integer r and any positive integer s where \(\gcd \left( p,s\right) =1.\) Moreover, we will provide examples of the number of these codes with different lengths \(n=r+s\).

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Acknowledgements

The authors would like to thank the reviewers for their careful reading of the paper and their valuable comments that improved the paper tremendously. In fact, the reviewers comments played a huge impact to create Sect. 5 of the paper that generalizes our results from counting the number of \({ {\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes to counting the numbers of \({{\mathbb {Z}}_{p}{{\mathbb {Z}}_{p^{2}}}}\)-additive cyclic codes.

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Correspondence to Ismail Aydogdu.

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Yildiz, E., Abualrub, T. & Aydogdu, I. On the number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\) and \({{\mathbb {Z}}}_{p}{{\mathbb {Z}}}_{p^{2}}\)-additive cyclic codes. AAECC 34, 81–97 (2023). https://doi.org/10.1007/s00200-020-00474-4

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  • DOI: https://doi.org/10.1007/s00200-020-00474-4

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