Let $\mathbb F_{p^m}$ be a finite field with $p^m$ elements, where $p$ is a prime and $m$ is an integer number. Let $R$ be the commutative ring $\mathbb F_{p^m}+u\mathbb F_{p^m}+u^2\mathbb F_{p^m}=\{a+bu+cu^2|a, b, c\in \mathbb F_{p^m}\}$ with $u^3=0$. The ring $R$ is a chain ring, which has a unique maximal ideal $\langle u\rangle=\{au|a\in \mathbb F_{p^m}\}$ (see [3]). A code of length $n$ over $R$ is a nonempty subset of $R^n$, and a code is linear over $R$ if it is an $R-$submodule of $R^n$. Let $C$ be a code of length $n$ over $R$ and $P(C)$ be its polynomial representation, i.e.,

$ P(C)=\{\sum\limits_{i=0}^{n-1}{{{c}_{i}}}{{x}^{i}}|({{c}_{0}}, {{c}_{1}}, \cdots, {{c}_{n-1}})\in C\}. $

The notions of cyclic shift and cyclic codes are standard for codes over $R$. Briefly, for the ring $R$, a cyclic shift on $R^n$ is a permutation $T$ such that

$ T(c_{0}, c_{1}, \cdots, c_{n-1})=(c_{n-1}, c_{0}, \cdots, c_{n-2}). $

A linear code over ring $R$ of length $n$ is cyclic if it is invariant under cyclic shift. It is known that a linear code over ring $R$ is cyclic if and only if $P(C)$ is an ideal of $\frac{R[x]}{\left\langle \left. {{x}^{n}}-1 \right\rangle \right.}$ (see [5]).

The following two theorems can be found in [1].

Theorem 1.1

Type 1   $\langle0\rangle, \langle1\rangle$.

Type 2   $I=\langle u(x-1)^i\rangle$, where $0\leq i \leq p^s-1$.

Type 3   $I=\langle(x-1)^i+u \sum\limits_{j=0}^{i-1}c_{1j}(x-1)^j\rangle, $ where $1\leq i \leq p^s-1, c_{1j}\in \mathbb F_{p^{m}}$; or equivalently, $I=\langle(x-1)^i+u(x-1)^th(x)\rangle$, where $1\leq i \leq p^s-1, 0\leq t< i$, and either $h(x)$ is 0 or $h(x)$ is a unit where it can be represented as $h(x)=\sum\limits_{j}h_j(x-1)^j$ with $h_j \in \mathbb F_{p^m}$, and $h_0\neq 0$.

Type 4   $I=\langle(x-1)^i+u\sum\limits_{j=0}^{w-1}c_{1j}(x-1)^j, u(x-1)^w\rangle$, where $1\leq i \leq p^s-1, c_{1j}\in\mathbb F_{p^m}, w<l$ and $w<T$, where $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u\sum\limits_{j=0}^{i-1}c_{1j}(x-1)^j\rangle$; or equivalently, $\langle(x-1)^i+u(x-1)^th(x), u(x-1)^w\rangle$, with $h(x)$ as in Type 3, and ${\rm deg}(h)\leq w-t-1$.

Theorem 1.2  Let $C$ be a cyclic code of length $p^s$ over $\mathbb F_{p^m}+u\mathbb F_{p^m}$, as classified in Theorem 1.1. Then the number of codewords $n_C$ of $\mathit{C}$ is determined as follows.

If $C = \langle0\rangle$, then $n_C=1$.

If $C = \langle1\rangle$, then $n_C=p^{2mp^s}$.

If $C = \langle u(x-1)^{i}\rangle$, where $ 0\leq i \leq p^s-1$, then $n_C=p^{m(p^s-i)}$.

If $C = \langle(x-1)^{i}\rangle$, where $ 1\leq i \leq p^s-1$, then $n_C=p^{2m(p^s-i)}$.

If $C = \langle(x-1)^{i}+u(x-1)^th(x)\rangle$, where $ 1\leq i \leq p^s-1, ~0\leq t< i$, and $h(x)$ is a unit, then

$ n_C=\left\{\begin{aligned} &p^{2m(p^s-i)}, ~~ ~~~~~~~{\rm if}~ 1\leq i\leq p^{s-1}+\frac{t}{2}, \\ &p^{m(p^s-t)}, ~~~~~ ~~~~~{\rm if}~ p^{s-1}+\frac{t}{2}< i \leq p^{s-1}-1. \end{aligned}\right. $

If $C = \langle(x-1)^{i}+u(x-1)^th(x), u(x-1)^{\kappa}\rangle$, where $ 1\leq i \leq p^s-1, ~0\leq t< i$, either $h(x)$ is 0 or $h(x)$ is a unit, and

$ \kappa<T=\left\{\begin{aligned} &i, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{\rm if}~ h(x)=0~, \\ &\min\{i, p^s-i+t\}, ~~~~~ ~~~~~&{\rm if}~ h(x)\neq 0, \end{aligned}\right. $

then $n_C=p^{m(2p^s-i-\kappa)}$.

Recently, Liu and Xu [3] studied constacyclic codes of length $p^s$ over $R$. In particular, they classified all cyclic codes of length $p^s$ over $R$. But they did not give the number of codewords in each of cyclic codes of length $p^s$ over $R$. In this note, we study repeated-root cyclic codes over $R$ by using the different method from [2], and obtain the number of codewords in each of cyclic codes of length $p^s$ over $R$.

Cyclic codes of length $p^s$ over $R$ are ideals of the residue ring $R_1=\frac{R[x]}{\langle x^{p^s}-1\rangle}$. It is easy to prove the ring $R_1$ is a local ring with the maximal ideal $\langle u, x-1\rangle$, but it is not a chain ring.

We can list all cyclic codes of length $p^s$ over $R_1$ as follows.

Theorem 2.1  Cyclic codes of length $p^s$ over $R$ are

Type 1   $\langle0\rangle, \langle1\rangle$.

Type 2   $I=\langle u^2(x-1)^k\rangle$, where $0\leq k \leq p^s-1$.

Type 3   $I=\langle u(x-1)^l+u^2\sum\limits_{j=0}^{l}c_{2j}(x-1)^j\rangle, $ where $0\leq l \leq p^s-1, c_{2j}\in \mathbb F_{p^{m}}$; or equivalently, $I=\langle u(x-1)^l+u^2(x-1)^th(x)\rangle$, where $0\leq l \leq p^s-1, 0\leq t< l$, and either $h(x)$ is 0 or $h(x)$ is a unit where it can be represented as $h(x)=\sum_{j}h_j(x-1)^j$ with $h_j \in \mathbb F_{p^m}$, and $h_0\neq 0$.

Type 4   $I=\langle u(x-1)^l+u^2\sum\limits_{j=0}^{w}c_{2j}(x-1)^j, u^2(x-1)^w\rangle$, where $1\leq l \leq p^s-1, c_{2j}\in \mathbb F_{p^m}, w<l$ and $w$ is the smallest integer such that $u^2(x-1)^w\in \langle u(x-1)^l+u^2\sum\limits_{j=0}^{l-1}c_{2j}(x-1)^j\rangle$; or equivalently, $I=\langle u(x-1)^l+u^2(x-1)^th(x), u(x-1)^w\rangle$, with $h(x)$ as in Type 3, and $deg(h)\leq w-t-1$.

Type 5   $I=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x)\rangle$, where $1\leq i\leq p^s-1, 0\leq t < i, 0\leq z < i$ and $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3.

Type 6   $I=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u^2(x-1)^\eta\rangle$, where $1\leq i\leq p^s-1, 0\leq t < i, 0\leq z < i$, $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3, $\eta< i$, and $\eta$ is the smallest integer such that $u^2(x-1)^{\eta}\in \langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x)\rangle$.

Type 7   $I=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u(x-1)^q+u^2\sum\limits_{j=0}^{q}e_{2j}(x-1)^j\rangle$, where $1\leq i\leq p^s-1, ~0\leq t\leq i, ~0\leq z\leq i, ~q<T\leq i$, $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u(x-1)^th_1(x)\rangle$, and $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3.

Type 8   $I=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u(x-1)^q+u^2\sum\limits_{j=0}^{\sigma}e_{2j}(x-1)^j, u^2(x-1)^\sigma\rangle$, where $1\leq i\leq p^s-1, \sigma< q \leq i, ~0\leq t\leq i, ~0\leq z\leq i, ~q< T \leq i$, $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u(x-1)^th_1(x)\rangle$, and $\sigma$ is the smallest integer such that $u^2(x-1)^{\sigma}\in \langle u(x-1)^q+u^2\sum\limits_{j=0}^{q-1}e_{2j}(x-1)^j\rangle$, and $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3.

Proof  Ideals of Type 1 are the trivial ideals. Consider an arbitrary nontrivial ideal of $R_1$.

Start with the homomorphism $\varphi:\mathbb F_{p^m}+u\mathbb F_{p^m}+u^2\mathbb F_{p^m}\rightarrow\mathbb F_{p^m}+u\mathbb F_{p^m}$ with $\varphi(a+ub+u^2c)=a+ub$. This homomorphism then can be extended to a homomorphism of rings of polynomials

$ \varphi:~R_1=\frac{(\mathbb F_{p^m}+u\mathbb F_{p^m}+u^2\mathbb F_{p^m})[x]}{\langle x^p-1\rangle}\rightarrow \overline{R_1}=\frac{(\mathbb F_{p^m}+u\mathbb F_{p^m})[x]}{\langle x^p-1\rangle} $

by letting $\varphi(c_{0}+c_{1}x+\cdots+c_{p^s-1}x^{p^s-1})=\varphi(c_{0})+\varphi(c_{1})x+\cdots+\varphi(c_{p^s-1})x^{p^s-1}.$ Note that Ker $\varphi=u^2\frac{\mathbb F_{p^m}[x]}{\langle x^{p^s}-1\rangle}$.

Now, let us assume that $I$ is a nontrivial ideal of $R_1$. Then $\varphi(I)$ is an ideal of $\overline{R_1}$. But ideals of $\overline{R_1}$ are characterized. So we can make use of these results.

On the other hand Ker $\varphi$ is also an ideal of $u^2\frac{\mathbb F_{p^m}[x]}{\langle x^{p^s}-1\rangle}$. We can consider it to be $u^2$ times a ideal of $\frac{\mathbb F_{p^m}[x]}{\langle x^{p^s}-1\rangle}$. This means that we can again use the results in the aforementioned papers. By using the characterization in [2], we have

$ {\rm Ker}\varphi=0~~{\rm or~~ Ker}\varphi=\langle u^2(x-1)^k\rangle, ~~0\leq k \leq p^s. $

For $\varphi(I)$, by using the characterization in [1], we shall discuss $\varphi(I)$ by carrying out the following cases.

Case 1   $\varphi(I)=0$. Then $I=\langle u^2(x-1)^k\rangle$, where $0\leq k \leq p^s-1$.

Case 2   $\varphi(I)\neq 0$.We now have seven subcases.

Case 2a   $\varphi(I)=\langle u(x-1)^l\rangle$, where $0\leq l \leq p^s-1$.

If Ker $\varphi=0$, then $I=\langle u(x-1)^l+u^2\sum\limits_{j=0}^{l}c_{2j}(x-1)^j\rangle$, where $0\leq l \leq p^s-1$, $ c_{2j}\in \mathbb F_{p^m}$, or equivalently, $I=\langle u(x-1)^l+u^2(x-1)^th(x)\rangle$, where $0\leq l \leq p^s-1, 0\leq t< l$, and either $h(x)$ is 0 or $h(x)$ is a unit where it can be represented as $h(x)=\sum_{j}h_j(x-1)^j$ with $h_j \in \mathbb F_{p^m}$, and $h_0\neq 0$.

If Ker $\varphi \neq 0$, then Ker $\varphi= \langle u^2(x-1)^w\rangle$, where $0\leq w \leq p^s-1$. Hence

$ I=\langle u{{(x-1)}^{l}}+{{u}^{2}}\sum\limits_{j=0}^{w}{{{c}_{2j}}}{{(x-1)}^{j}}, {{u}^{2}}{{(x-1)}^{w}}\rangle, $

where $1\leq l \leq p^s-1$, $c_{2j}\in \mathbb F_{p^m}$, $w<l$ and $w$ is the smallest integer such that $u^2(x-1)^w \in \langle u(x-1)^l+u^2\sum\limits_{j=0}^{l-1}c_{2j}(x-1)^j\rangle$, or equivalently, $\langle u(x-1)^l+u^2(x-1)^th(x), u(x-1)^w\rangle$, with $h(x)$ as in Type 3, and $\rm{deg}(\mathit{h})\le \mathit{w}-\mathit{t}-1$.

Case 2b   $\varphi(I)=\langle(x-1)^i+u\sum\limits_{j=0}^{i-1}c_{2j}(x-1)^j\rangle=\langle(x-1)^i+u(x-1)^th_1(x)\rangle$, where $1\leq i \leq p^s-1$, $ c_{2j}\in \mathbb F_{p^m}$, and $h_{1}(x)$ as in Type 3.

If Ker $\varphi=0$, then $I=\langle(x-1)^i+u\sum\limits_{j=0}^{i-1}c_{1j}(x-1)^j+u^2\sum\limits_{j=0}^{i-1}c_{2j}(x-1)^j\rangle=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x)\rangle$, where $1\leq i\leq p^s-1$, $ c_{1j}, c_{2j}\in \mathbb F_{p^m}$, $0\leq t < i, 0\leq z < i$, and $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3.

If Ker $\varphi \neq 0$, then

$ I=\langle {{(x-1)}^{i}}+u\sum\limits_{j=0}^{i-1}{{{c}_{1j}}}{{(x-1)}^{j}}+{{u}^{2}}\sum\limits_{j=0}^{\eta }{{{c}_{2j}}}{{(x-1)}^{j}}, {{u}^{2}}{{(x-1)}^{\eta }}\rangle $

or

$ I=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u^2(x-1)^\eta\rangle, $

where $1\leq i \leq p^s-1$, $ c_{1j}, c_{2j}\in \mathbb F_{p^m}$, $\eta<i$, $\eta$ is the smallest integer such that $u^2(x-1)^{\eta}\in \langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x)\rangle$, and $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3.

Case 2c   $\varphi(I)=\langle(x-1)^i+u(x-1)^th_{1}(x), u(x-1)^q\rangle$, where $1\leq i \leq p^s-1, ~0\leq t\leq i, ~q<T$, and $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u(x-1)^th_1(x)\rangle$, $h_1(x)$ is similar to $h(x)$ in Type 3.

If Ker $\varphi=0$, then $I=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u(x-1)^q+u^2\sum\limits_{j=0}^{q-1}e_{2j}(x-1)^j\rangle$, where $1\leq i\leq p^s-1, ~0\leq t\leq i, ~0\leq z\leq i, ~ q<T\leq i$, $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u(x-1)^th_1(x)\rangle$, and $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3.

If Ker $\varphi \neq 0$, then $I=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u(x-1)^q+u^2\sum\limits_{j=0}^{\sigma}e_{2j}(x-1)^j, u^2(x-1)^\sigma\rangle$, where $1\leq i\leq p^s-1, ~0\leq t\leq i, ~0\leq z\leq i, ~\sigma< q \leq i$, $q< T \leq i$, $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u(x-1)^th_1(x)\rangle$, and $\sigma$ is the smallest integer such that $u^2(x-1)^{\sigma}\in \langle u(x-1)^q+u^2\sum\limits_{j=0}^{q}e_{2j}(x-1)^j\rangle$, and $h_1(x), h_2(x)$ are similar to $h(x)$ in Type 3.

By Theorem 6.2 in [2], each cyclic code of length $p^s$ over $\mathbb F_{p^m}$ is an ideal of the form $\langle(x-1)^i\rangle$ of the chain ring $\frac{\mathbb F_{p^m}[x]}{\langle x^{p^s}-1\rangle}$, where $0\leq i\leq p^s$, and this code $\langle(x-1)^i\rangle$ contains $p^{m(p^s-i)}$ codewords. In light of Theorem 1.2, we can now determine the sizes of all cyclic codes of length $p^s$ over $R$ by multiplying the sizes of $\varphi(C)$ and Ker $\varphi$ in each case.

Theorem 2.2  Let $C$ be a cyclic code of length $p^s$ over $R$, as classified in Theorem 2.1. Then the number of codewords $n_C$ of $C$ is determined as follows.

If $C=\langle0\rangle$, then $n_C=1$.

If $C=\langle1\rangle$, then $n_C=p^{3mp^s}$.

If $C=\langle u^2(x-1)^k\rangle$, where $0\leq k \leq p^s-1$, then $n_C=p^{m(p^s-k)}$.

If $C=\langle u(x-1)^l+u^2\sum\limits_{j=0}^{l}c_{2j}(x-1)^j\rangle, $ where $0\leq l \leq p^s-1, c_{2j}\in \mathbb F_{p^{m}}$, then $n_C=p^{m(p^s-l)}$.

If $C=\langle u(x-1)^l+u^2\sum\limits_{j=0}^{w}c_{2j}(x-1)^j, u^2(x-1)^w\rangle$, where $0\leq l \leq p^s-1, c_{2j}\in \mathbb F_{p^m}, w<l$ and $w$ the smallest integer such that $u^2(x-1)^w\in \langle u(x-1)^l+u^2\sum\limits_{j=0}^{l-1}c_{2j}(x-1)^j\rangle$, then $n_C=p^{2mp^s-m(l+w)}$.

If $C=\langle(x-1)^i\rangle$, where $1\leq i \leq p^s-1$, then $n_C=p^{2m(p^s-i)}$.

If $C=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x)\rangle$, where $1\leq i\leq p^s-1, 0\leq t < i, 0\leq z < i$ and $h_1(x)$ is a unit, then

$ n_C=\left\{\begin{aligned} &p^{2m(p^s-i)}, ~~ ~~~~~~~{\rm if}~ 1\leq i\leq p^{s-1}+\frac{t}{2}, \\ &p^{m(p^s-t)}, ~~~~~ ~~~~~{\rm if}~ p^{s-1}+\frac{t}{2}< i \leq p^{s-1}-1. \end{aligned}\right. $

If $C=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u^2(x-1)^\eta\rangle$, where $1\leq i\leq p^s-1, 0\leq t < i, 0\leq z < i$, $h_1(x)$ is a unit, $\eta< i$, $\eta$ is the smallest integer such that $u^2(x-1)^{\eta}\in \langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x)\rangle$, and $h_1(x)$ is a unit, then

$ n_C=\left\{\begin{aligned} &p^{3mp^s-2mi-m\eta}, ~~ ~~~~~~~{\rm if}~ 1\leq i\leq p^{s-1}+\frac{t}{2}, \\ &p^{2mp^s-m(t+\eta)}, ~~~~~ ~~~~~{\rm if}~ p^{s-1}+\frac{t}{2}< i \leq p^{s-1}-1. \end{aligned}\right. $

If $C=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u(x-1)^q+u^2\sum\limits_{j=0}^{q}e_{2j}(x-1)^j\rangle$, where $1\leq i\leq p^s-1, q<T\leq i$, $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u(x-1)^th_1(x)\rangle$, either $h_1(x), h_2(x)$ are 0 or $h_1(x), h_2(x)$ are units, and

$ q<T=\left\{\begin{aligned} &i, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{\rm if}~ h_{1}(x)=0~, \\ &\min\{i, p^s-i+t\}, ~~~~~ ~~~~~&{\rm if}~ h_{1}(x)\neq 0, \end{aligned}\right. $

then $n_C=p^{m(2p^s-i-q)}$.

If $C=\langle(x-1)^i+u(x-1)^th_1(x)+u^2(x-1)^zh_2(x), u(x-1)^q+u^2\sum\limits_{j=0}^{\sigma}e_{2j}(x-1)^j, u^2(x-1)^\sigma\rangle$, where $1\leq i\leq p^s-1, \sigma< q \leq i$, $q< T \leq i$, $T$ is the smallest integer such that $u(x-1)^T\in \langle(x-1)^i+u(x-1)^th_1(x)\rangle$, and $\sigma$ is the smallest integer such that $u^2(x-1)^{\sigma}\in \langle u(x-1)^q+u^2\sum\limits_{j=0}^{q}e_{2j}(x-1)^j\rangle$, either $h_1(x), h_2(x)$ are 0 or $h_1(x), h_2(x)$ are units, and

$ q<T=\left\{\begin{aligned} &i, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~&{\rm if}~ h_{1}(x)=0, \\ &\min\{i, p^s-i+t\}, ~~~~~ ~~~~~&{\rm if}~ h_{1}(x)\neq 0, \end{aligned}\right. $

then $n_C=p^{3mp^s-m(i+q+\sigma)}$.