### Cyclic codes of length $5p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$ and their duals

#### Abstract

For an odd prime $p\not=5$, the structures of cyclic codes of length $5p^s$ over $\mathcal R=\mathbb F_{p^m}+u\mathbb F_{p^m} (u^2=0)$ are completely determined. Cyclic codes of length $5p^s$ over $\mathcal R$ are considered in 3 cases, namely, $p\equiv 1 \pmod 5$, $p\equiv 4\pmod 5$, $p\equiv 2 ~~{\text or}~~ 3\pmod 5$. When $p\equiv 1\pmod 5$, a cyclic code of length $5p^s$ over $\mathcal R$ can be expressed as a direct sum of a cyclic code and $\gamma_i^{p^s}$-constacyclic codes of length $p^s$ over $\mathcal R$, where $\gamma_i^{p^s}=-\frac{i(p^m-1)p^s}{10}, i=1,3,7,9$. When $p \equiv 4\pmod 5$, it is equivalent to $p^m\equiv 1\pmod 5$ when $m$ is even and $p^m\equiv 4 \pmod 5$ when $m$ is odd. If $p^m\equiv 1 \pmod 5$ when $m$ is even, then a cyclic code of length $5p^s$ over $\mathcal R$ can be obtained as a direct sum of a cyclic code and $\gamma_i^{p^s}$-constacyclic codes of length $p^s$ over $\mathcal R$, where $\gamma_i^{p^s}=-\frac{i(p^m-1)p^s}{10}, i=1,3,7,9$. If $p^m\equiv 4 \pmod 5$ when $m$ is odd, then a cyclic code of length $5p^s$ over $\mathcal R$ can be expressed as a direct sum of a cyclic code of length $p^s$ over $\mathcal R$ and an $\alpha_1$ and $\alpha_2$-constacyclic code of length $2p^s$ over $\mathcal R$, for some $\alpha_1, \alpha_2\in \mathbb F_{p^m}\setminus \{0\}$. If $p\equiv 2 ~~{\text or}~~ 3\pmod 5$ such that $p^m\not\equiv 1 \pmod 5$, then a cyclic code of length $5p^s$ over $\mathcal R$ can be expressed as $C_1\oplus C_2$, where $C_1$ is an ideal of $\frac{\mathcal R[x]}{\langle x^{p^s}-1\rangle}$ and $C_2$ is an ideal of $\frac{\mathcal R[x]}{\langle (x^4+x^3+x^2+x+1)^{p^s}\rangle}$. We also investigate all ideals of $\frac{\mathcal R[x]}{\langle (x^4+x^3+x^2+x+1)^{p^s}\rangle}$ to study detail structure of a cyclic code of length $5p^s$ over $\mathcal R$.

In addition, dual codes of all cyclic codes of length $5p^s$ over $\mathcal R$ are also given.

Furthermore, we give the number of codewords in each of those cyclic codes of length $5p^s$ over $\mathcal R$. As cyclic and negacyclic codes of length $5p^s$ over $\mathcal R$ are in a one-by-one equivalent via the ring isomorphism $x \mapsto -x$, all our results for cyclic codes hold true accordingly to negacyclic codes.

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