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

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 $$n=r+s$$ .