Using the polynomial representation of codewords in $\begin{array}{}{R}_{k}^{n}\end{array}$, we easily have the following.

#### Lemma 3.1

*A subset* *C* *of* $\begin{array}{}{R}_{k}^{n}\end{array}$ *is* *a* λ-*constacyclic code of length* *n* *over* *R*_{k} *if and only if its polynomial representation is an ideal of the ring* *R*_{k}[*x*]/〈*x*^{n} - λ〉.

For any *r* = (*r*^{(0)}, *r*^{(1)}, ⋯, *r*^{(n−1)}) ∈ $\begin{array}{}{R}_{k}^{n}\end{array}$, where *r*^{(i)} = $\begin{array}{}\sum _{j=1}^{{2}^{k}}\end{array}$ *r*_{ij}*e*_{j}, *i* = 0, 1, ⋯, *n* − 1. Then *r* can be uniquely express as *r* = $\begin{array}{}\sum _{j=1}^{{2}^{k}}\end{array}$ *r*_{j}*e*_{j}, where *r*_{j} = (*r*_{0j}, *r*_{1j}, ⋯, *r*_{n−1,j}) ∈ $\begin{array}{}{\mathbb{F}}_{{p}^{m}}^{n},\end{array}$ *j* = 1, 2,⋯, 2^{k}.

For any *r*, *s* ∈ $\begin{array}{}{R}_{k}^{n}\end{array}$, where *s* = $\begin{array}{}\sum _{j=1}^{{2}^{k}}\end{array}$ *s*_{j}*e*_{j}, *s*_{j} = (*s*_{0j}, *s*_{1j}, ⋯, *s*_{n−1,j}) ∈ $\begin{array}{}{\mathbb{F}}_{{p}^{m}}^{n},\end{array}$ we can get that

$$\begin{array}{}{\displaystyle r\cdot s=\sum _{j=1}^{{2}^{k}}({r}_{j}\cdot {s}_{j}){e}_{j},}\end{array}$$

where $\begin{array}{}{r}_{j}\cdot {s}_{j}=\sum _{i=0}^{n-1}({r}_{ij}{s}_{ij}).\end{array}$

Let *C* be a linear code over *R*_{k}. For *j* = 1, 2,⋯, 2^{k}, we denote *C*_{j} as follows:

$$\begin{array}{}{\displaystyle {C}_{j}=\{{r}_{j}\in {\mathbb{F}}_{{p}^{m}}^{n}|\sum _{i=1}^{{2}^{k}}{r}_{i}{e}_{i}\in C,{r}_{i}\in {\mathbb{F}}_{{p}^{m}}^{n},\},j=1,2,\cdots ,{2}^{k}.}\end{array}$$

Clearly, *C*_{j} is a linear code of length *n* over 𝔽_{pm}.

By the definition above we have the following theorems easily.

#### Theorem 3.2

*Let* *C* *be a linear code over* *R*_{k}, *then* $\begin{array}{}C=\sum _{j=1}^{{2}^{k}}{e}_{j}{C}_{j},|C|=\prod _{j=1}^{{2}^{k}}|{C}_{j}|,\end{array}$ *where* *C*_{1}, *C*_{2}, ⋯, *C*_{2k} *are linear codes of length* *n* *over* 𝔽_{pm}, *and the direct sum decomposition is unique*.

#### Theorem 3.3

*Let* *C* *be a linear code over* *R*_{k}, *then* $\begin{array}{}{C}^{\mathrm{\perp}}=\sum _{j=1}^{{2}^{k}}{e}_{j}{C}_{j}^{\mathrm{\perp}},\phantom{\rule{thinmathspace}{0ex}}where\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{C}_{j}^{\mathrm{\perp}}\end{array}$ *is the dual code of* *C*_{j}, *where* *j* = 1, 2,⋯, 2^{k}.

#### Proof

Let $\begin{array}{}\stackrel{~}{C}=\sum _{j=1}^{{2}^{k}}{e}_{j}{C}_{j}^{\mathrm{\perp}}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}For any}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}c\in C,\stackrel{~}{c}\in \stackrel{~}{C},c\cdot \stackrel{~}{c}=\sum _{j=1}^{{2}^{k}}({c}_{j}\stackrel{~}{{c}_{j}}){e}_{j},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}where}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}c=\sum _{j=1}^{{2}^{k}}{e}_{j}{c}_{j},\stackrel{~}{c}=\sum _{j=1}^{{2}^{k}}{e}_{j}\stackrel{~}{{c}_{j}},{c}_{j}\in {C}_{j},\stackrel{~}{{c}_{j}}\in {C}_{j}^{\mathrm{\perp}}.\end{array}$ Then *c* ⋅ *c̃* = 0, and thus *C̃* ⊆ *C*^{⊥}. The ring *R*_{k} is a principal ideal ring and thus a Frobenius ring, we have |*C*||*C*^{⊥}| = |*R*_{k}|^{n}. Thus

$$\begin{array}{}{\displaystyle |\stackrel{~}{C}|=\prod _{j=1}^{{2}^{k}}|{C}_{j}^{\mathrm{\perp}}|=\prod _{j=1}^{{2}^{k}}\frac{{p}^{n}}{|{C}_{j}|}=\frac{|{R}_{k}{|}^{n}}{|C|}=|{C}^{\mathrm{\perp}}|.}\end{array}$$

So *C*^{⊥} = *C̃*. □

#### Theorem 3.4

*Let* *C* *be a linear code over* *R*_{k}, *then* *C* *is a self*-*orthogonal code if and only if* *C*_{j} *is a self*-*orthogonal over* 𝔽_{pm}, *where* $\begin{array}{}c=\sum _{j=1}^{{2}^{k}}{e}_{j}{c}_{j}.\end{array}$ *C* *is a self*-*dual code if and only if* *C*_{j} *is a self*-*dual code over* 𝔽_{pm}, *where* *j* = 1, 2,⋯, 2^{k}.

#### Proof

By Theorems 3.2 and 3.3, *C* ⊆ *C*^{⊥} if and only if *C*_{j} ⊆ $\begin{array}{}{C}_{j}^{\mathrm{\perp}},\end{array}$ so if *C* is a self-orthogonal code then *C*_{j} is a self-orthogonal code over 𝔽_{pm}, where *j* = 1, 2,⋯, 2^{k}. Similarly, *C* is a self-dual code then *C*_{j} is a self-dual code over 𝔽_{pm}, where *j* = 1, 2,⋯, 2^{k}. □

Let *C* be a linear code of length *n* over *R*_{k}, for any *c* = *c*_{1}*e*_{1} + *c*_{2}*e*_{2} + ⋯ + *c*_{2k}*e*_{2k} ∈ *C*, *Φ*(*c*) = (*c*_{1}, *c*_{2}, ⋯, *c*_{2k}) ∈ $\begin{array}{}{\mathbb{F}}_{{p}^{m}}^{{2}^{k}n}\end{array}$. Let *C*_{1},*C*_{2}, ⋯, *C*_{2k} be linear codes of length *n* over 𝔽_{pm}, we define

$$\begin{array}{}{\displaystyle {C}_{1}\times {C}_{2}\times \cdots \times {C}_{{2}^{k}}=\{({c}_{1},{c}_{2},\cdots ,{c}_{{2}^{k}}),{c}_{i}\in {C}_{i},i=1,2,\cdots ,{2}^{k}\}.}\end{array}$$

#### Theorem 3.5

*Let* *C* = *e*_{1}*C*_{1} + *e*_{2}*C*_{2} + ⋯ + *e*_{2k}*C*_{2k} *be a linear code of length* *n* over *R*_{k} *with* |*C*| = *p*^{ml} *and the minimum Lee distance* *d*_{L}(*C*) = *d*. *Then* *Φ*(*C*) = *C*_{1}× *C*_{2}× ⋯ × *C*_{2k} *is a linear code with parameter* [2^{k}*n*, *l*, *d*] *and* *Φ*(*C*)^{⊥} = *Φ*(*C*^{⊥}). *If* *C* *is a self*-*dual code over* *R*_{k}, *then* *Φ*(*C*) *is a self*-*dual code over* 𝔽_{pm}.

#### Proof

By the definition above, we can know that

$$\begin{array}{}{\displaystyle {C}_{1}\times {C}_{2}\times \cdots \times {C}_{{2}^{k}}\subseteq \mathit{\Phi}(C)}\end{array}$$

and

$$\begin{array}{}{\displaystyle |{C}_{1}\times {C}_{2}\times \cdots \times {C}_{{2}^{k}}|=|{C}_{1}||{C}_{2}|\cdots |{C}_{{2}^{k}}|=|C|.}\end{array}$$

This gives that

$$\begin{array}{}{\displaystyle \mathit{\Phi}(C)={C}_{1}\times {C}_{2}\times \cdots \times {C}_{{2}^{k}}.}\end{array}$$

Let $\begin{array}{}c=\sum _{j=1}^{{2}^{k}}{e}_{j}{c}_{j}\in C,d=\sum _{j=1}^{{2}^{k}}{e}_{j}{d}_{j}\in {C}^{\perp},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{where}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{c}_{j}\in {C}_{j},{d}_{j}\in {C}_{j}^{\perp},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{then}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}c\cdot d=\sum _{j=1}^{{2}^{k}}{e}_{j}{c}_{j}{d}_{j}=0,\end{array}$ which implies *c*_{j}*d*_{j} = 0, so

$$\begin{array}{}{\displaystyle \mathit{\Phi}(c)\cdot \mathit{\Phi}(d)=\sum _{j=1}^{{2}^{k}}{c}_{j}{d}_{j}=0,}\end{array}$$

which implies

$$\begin{array}{}{\displaystyle \mathit{\Phi}(C{)}^{\mathrm{\perp}}\supseteq \mathit{\Phi}({C}^{\perp}).}\end{array}$$

By Theorem 3.3, we have

$$\begin{array}{}{\displaystyle \mathit{\Phi}({C}^{\perp})={C}_{1}^{\perp}\times {C}_{2}^{\perp}\times \cdots \times {C}_{{2}^{k}}^{\perp}.}\end{array}$$

Since *Φ* is one-to-one, we have

$$\begin{array}{}{\displaystyle |\mathit{\Phi}({C}^{\perp})|=\frac{{p}^{m{2}^{k}n}}{|C|}=\frac{{p}^{m{2}^{k}n}}{|\mathit{\Phi}(C)|}=|\mathit{\Phi}(C{)}^{\mathrm{\perp}}|.}\end{array}$$

So

$$\begin{array}{}{\displaystyle \mathit{\Phi}(C{)}^{\mathrm{\perp}}=\mathit{\Phi}({C}^{\mathrm{\perp}}).}\end{array}$$ □

Let *τ* be a cyclic shift operator on $\begin{array}{}{\mathbb{F}}_{{p}^{m}}^{n}.\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{Let}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}a=({a}^{(1)}|{a}^{(2)}|\cdots |{a}^{({2}^{k})})\in {\mathbb{F}}_{{p}^{m}}^{{2}^{k}n},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{\hspace{0.17em}where}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{a}^{(j)}\in {\mathbb{F}}_{{p}^{m}}^{n}\end{array}$ for *j* = 1, 2,⋯, 2^{k}. Let *τ*_{2k} be the quasi-shift given by

$$\begin{array}{}{\displaystyle {\tau}_{{2}^{k}}({a}^{(1)}|{a}^{(2)}|\cdots |{a}^{({2}^{k})})=(\tau ({a}^{(1)})|\tau ({a}^{(2)})|\cdots |\tau ({a}^{({2}^{k})})).}\end{array}$$

#### Proposition 3.6

*Let* *σ* *be a cyclic shift on* $\begin{array}{}{R}_{k}^{n}\end{array}$, *let* *Φ* *be the Gray map from* $\begin{array}{}{R}_{k}^{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}to\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbb{F}}_{{p}^{m}}^{{2}^{k}n}\end{array}$, *and let* *τ*_{2k} *be as above*. *Then* *Φ**σ* = *τ*_{2k}*Φ*.

#### Proof

Let *r* = (*r*_{0}, *r*_{1}, ⋯, *r*_{n−1}) ∈ $\begin{array}{}{R}_{k}^{n}\end{array}$, where $\begin{array}{}{r}_{i}=\sum _{j=1}^{{2}^{k}}\end{array}$ *r*_{ij}*e*_{j}, *i* = 0, 1, ⋯, *n* − 1. We have *σ*(*r*) = (*r*_{n−1}, *r*_{0}, ⋯, *r*_{n−2}). If we apply *Φ*, we have

$$\begin{array}{}{\displaystyle \mathit{\Phi}(\sigma (r))=\mathit{\Phi}({r}_{n-1},{r}_{0},\cdots ,{r}_{n-2})=({r}_{1,n-1},{r}_{1,0},\cdots ,{r}_{1,n-2},{r}_{2,n-1},{r}_{2,0},\cdots ,{r}_{2,n-2},\cdots ,{r}_{{2}^{k},n-1},{r}_{{2}^{k},0},\cdots ,{r}_{{2}^{k},n-2}).}\end{array}$$

On the other hand,

$$\begin{array}{}{\displaystyle {\tau}_{{2}^{k}}(\mathit{\Phi}(r))={\tau}_{{2}^{k}}(\mathit{\Phi}({r}_{0},{r}_{1},\cdots ,{r}_{n-1}))={\tau}_{{2}^{k}}({r}_{1,0},{r}_{1,1},\cdots ,{r}_{1,n-1},{r}_{2,0},{r}_{2,1},\cdots ,{r}_{2,n-1},\cdots ,{r}_{{2}^{k},0},{r}_{{2}^{k},1},\cdots ,{r}_{{2}^{k},n-1})}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=({r}_{1,n-1},{r}_{1,0},\cdots ,{r}_{1,n-2},{r}_{2,n-1},{r}_{2,0},\cdots ,{r}_{2,n-2},\cdots ,{r}_{{2}^{k},n-1},{r}_{{2}^{k},0},\cdots ,{r}_{{2}^{k},n-2}).\end{array}$$

Therefore, we have

$$\begin{array}{}{\displaystyle \mathit{\Phi}\sigma ={\tau}_{{2}^{k}}\mathit{\Phi}.}\end{array}$$ □

#### Theorem 3.7

*Let* *C* *be a cyclic code of length* *n* *over* *R*_{k}. *Then* *Φ*(*C*) *is a quasi*-*cyclic code of index* 2^{k} *over* 𝔽_{pm} *with length* 2^{k}*n*.

#### Proof

Since *C* is a cyclic code, then *σ*(*C*) = *C*. If we apply *Φ*, we have *Φ**σ*(*C*) = *Φ*(*C*). By the Proposition 3.6, *Φ*(*σ*(*C*)) = *Φ*(*C*) = *τ*_{2k}(*Φ*(*C*)), so *Φ*(*C*) is a quasi-cyclic code of index 2^{k} over 𝔽_{pm} with length 2^{k}*n*. □

Let *C* be a linear code of length *n* over *R*_{k}, let *A*_{0}, *A*_{1}, ⋯, *A*_{2kn} denote the number of codewords in *C* of the Lee weight, and the Lee weight distribution of *C* is simply the tuple of numbers {*A*_{0}, *A*_{1}, ⋯, *A*_{2kn}}.

Let Lee_{C}(*x*, *y*) = $\begin{array}{}\sum _{i=0}^{{2}^{k}n}{A}_{i}{x}^{{2}^{k}n-i}{y}^{i}\end{array}$ denote the Lee weight enumerator of *C*, we get that

$$\begin{array}{}{\displaystyle {\mathrm{L}\mathrm{e}\mathrm{e}}_{C}(x,y)=\sum _{c\in C}{x}^{{2}^{k}n-{w}_{L}(c)}{y}^{{w}_{L}(c)}.}\end{array}$$

Let *W*_{C}(*x*, *y*) = $\begin{array}{}\sum _{c\in C}{x}^{{2}^{k}n-{w}_{H}(c)}{y}^{{w}_{H}(c)}\end{array}$ denote the Hamming weight enumerator of *C*.

By the results of [22], we have

$$\begin{array}{}{\displaystyle {W}_{{C}^{\mathrm{\perp}}}(x,y)=\frac{1}{|C|}{W}_{C}(x+(|{R}_{k}|-1)y,x-y).}\end{array}$$

By a proof similar to (cf. [23, Lemma 1]), we obtain the following lemma.

#### Lemma 3.8

*Let* *x* *and* *y* *be two vectors in* $\begin{array}{}{R}_{k}^{n}\end{array}$, *and let* *d*_{H}(*Φ*(*x*), *Φ*(*y*)) *denote the Hamming distance of* *Φ*(*x*),*Φ*(*y*), *where* *Φ*(*x*), *Φ*(*y*) *are codewords in* $\begin{array}{}{\mathbb{F}}_{{p}^{m}}^{{2}^{k}n}\end{array}$. *Let* *w*_{H}(*Φ*(*x*)) *denote the Hamming weight of* *Φ*, *then*

*w*_{L}(*x*) = *w*_{H}(*Φ*(*x*)),

*d*_{L}(*x*, *y*) = *d*_{H}(*Φ*(*x*), *Φ*(*y*)).

#### Theorem 3.9

*Let* *C* *be a linear code of length* *n* *over* *R*_{k}, *then* Lee_{C⊥}(*x*, *y*) = $\begin{array}{}\frac{1}{|\mathit{\Phi}(C)|}\end{array}$ *W*_{Φ(C)}(*x* + (*p*^{m2k} − 1)*y*, *x* − *y*).

#### Proof

By Theorem 3.5, we have that

$$\begin{array}{}{\displaystyle {\mathrm{L}\mathrm{e}\mathrm{e}}_{{C}^{\mathrm{\perp}}}(x,y)={W}_{\mathit{\Phi}({C}^{\mathrm{\perp}})}(x,y)={W}_{\mathit{\Phi}(C{)}^{\mathrm{\perp}}}(x,y).}\end{array}$$

So

$$\begin{array}{}{\displaystyle {\mathrm{L}\mathrm{e}\mathrm{e}}_{C}(x,y)=\sum _{c\in C}{x}^{{2}^{k}n-{w}_{L}(c)}{y}^{{w}_{L}(c)}=\sum _{\mathit{\Phi}(c)\subseteq \mathit{\Phi}(C)}{x}^{{2}^{k}n-{w}_{H}(\mathit{\Phi}(c))}{y}^{{w}_{H}(\mathit{\Phi}(c))}={W}_{\mathit{\Phi}(C)}(x,y).}\end{array}$$

As *Φ* is one-to-one, we have that |*Φ*(*C*)| = |*C*|, hence

$$\begin{array}{}{\displaystyle {\mathrm{L}\mathrm{e}\mathrm{e}}_{{C}^{\mathrm{\perp}}}(x,y)={W}_{\mathit{\Phi}({C}^{\mathrm{\perp}})}(x,y)=\frac{1}{|\mathit{\Phi}(C)|}{W}_{\mathit{\Phi}(C)}(x+({p}^{m{2}^{k}}-1)y,x-y).}\end{array}$$ □

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