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Published in:
10-01-2021 | Original Paper
On the number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\) and \({{\mathbb {Z}}}_{p}{{\mathbb {Z}}}_{p^{2}}\)-additive cyclic codes
Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 1/2023
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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 \(n=r+s\).