Contraction of Cyclic Codes Over Finite Chain Rings

TL;DR

This paper studies the structure of cyclic codes over finite chain rings, presenting a trace representation and analyzing how contractions of these codes relate to constacyclic codes under certain conditions.

Contribution

It introduces a trace representation for cyclic codes over finite chain rings and characterizes contractions as constacyclic codes with specific parameters.

Findings

Trace representation of cyclic codes over finite chain rings

Contractions of cyclic codes yield constacyclic codes under gcd condition

Characterization of code length and structure relationships

Abstract

Let $\texttt{R}$ be a commutative finite chain ring of invariants $(q,s)$ and $\Gamma(\texttt{R})$ the Teichm\"uller's set of $\texttt{R}.$ In this paper, the trace representation cyclic $\texttt{R}$-linear codes of length $\ell,$ is presented, when $\texttt{gcd}(\ell, q) = 1.$ We will show that the contractions of some cyclic $\texttt{R}$-linear codes of length $u\ell$ are $\gamma$-constacyclic $\texttt{R}$-linear codes of length $\ell,$ where $\gamma\in\Gamma(\texttt{R})$ and the multiplicative order of is $u.$

Equations

$$\displaystyle G=\left(\begin{array}[]{cccccc}I_{k_{0}}&G_{0,1}&G_{0,2}&\cdots&G_{0,s-1}&G_{0,s}\\ 0&\theta I_{k_{1}}&\theta G_{1,2}&\cdots&\theta G_{1,s-1}&\theta G_{1,s}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&\cdots&\theta^{s-1}I_{k_{s-1}}&\theta^{s-1}G_{s-1,s}\end{array}\right)U,$$

$$\sigma (\mathcal{B}):=\left\{(\sigma ({\text{c}}_0),\cdots ,\sigma ({\text{c}}_{\ell -1})):({\text{c}}_0,\cdots ,{\text{c}}_{\ell -1})\in \mathcal{B}\right\}$$

$${\text{Tr}}_{\text{R}}^{\text{S}}(\mathcal{B}):=\left\{({\text{Tr}}_{\text{R}}^{\text{S}}({\text{c}}_0),\cdots ,{\text{Tr}}_{\text{R}}^{\text{S}}({\text{c}}_{\ell -1})):({\text{c}}_0,\cdots ,{\text{c}}_{\ell -1})\in \mathcal{B}\right\}.$$

$$u\mathrm{A}:=\{uz(\text{mod}\ell ):z\in \mathrm{A}\}.$$

$$\begin{array}[]{ll}\complement_{q}(\{1\})=\{1,3,9,7\};&\complement_{q}(\{2\})=\{2,6,18,14\};\\ \complement_{q}(\{4\})=\{4,12,16,8\};&\complement_{q}(\{11\})=\{11,13,19,17\}.\end{array}$$

$$\begin{array}[]{cccc}\textbf{ev}_{\eta}:&\textbf{P}(\texttt{S}\,;\,\mathrm{A})&\rightarrow&\texttt{S}^{\ell}\\ &f&\mapsto&(f(1),f(\eta),\cdots,f(\eta^{\ell-1})),\end{array}$$

$$\widetilde{{\text{L}}_{\eta }(\text{S};\mathrm{A})}=\sum _{i=0}^{m-1}{\text{L}}_{\eta }(\text{S};q^i\mathrm{A})={\text{L}}_{\eta }\left(\text{S};\stackrel{m-1}{\bigcup _{i=0}}{q}^i\mathrm{A}\right).$$

$${\text{Tr}}_{\text{R}}^{\text{S}}\circ {\text{ev}}_{\eta }(X^a):={\text{Tr}}_{\text{R}}^{\text{S}}\left(1,{\eta }^a,\cdots ,{\eta }^{a(\ell -1)}\right),$$

$${\text{rank}}_{\text{R}}({\text{C}}_{\eta }(\text{R};\mathrm{A}))={\text{rank}}_{\text{S}}({\text{L}}_{\eta }(\text{S};{\complement }_q(\mathrm{A})))=|{\complement }_q(\mathrm{A})|.$$

$$\begin{array}[]{cccc}\psi_{z}:&\texttt{R}[\xi^{m_{z}}]&\longrightarrow&\mathrm{\textbf{C}}_{\eta}\left(\texttt{R}\,;\,\{z\}\right)\\ &\textbf{a}&\longmapsto&\texttt{Tr}_{\texttt{R}}^{\texttt{S}}\left(\emph{ev}_{\eta}(\textbf{a}X^{z})\right)\\ \end{array}$$

$${\Re }_{\ell }(q,s):=\left\{({\mathrm{A}}_0,{\mathrm{A}}_1,\cdots ,{\mathrm{A}}_s):\left(\exists \lambda \in \{0,1,\cdots ,s{\}}^{{\Sigma }_{\ell }(q)}\right)\left({\mathrm{A}}_t={\lambda }^{-1}(\{t\})\right)\right\}$$

$$(|{\complement }_q({\mathrm{A}}_0)|,|{\complement }_q({\mathrm{A}}_1)|,\cdots ,|{\complement }_q({\mathrm{A}}_{s-1})|),$$

$$\mathcal{C}=\underset{z\in {\Sigma }_{\ell }(q)}{⨁}{\theta }^{t_z}{\text{C}}_{\eta }(\text{R};\{z\})=\stackrel{s-1}{\underset{t=0}{⨁}}{\theta }^t{\text{C}}_{\eta }(\text{R};{\mathrm{A}}_t),$$

$$\displaystyle\begin{array}[]{cccc}\wp:&\texttt{R}^{\ell}&\rightarrow&\texttt{R}^{u\ell}\\ &\underline{\textbf{c}}&\mapsto&(\gamma^{u-1}\underline{\textbf{c}}\;|\;\gamma^{u-2}\underline{\textbf{c}}\;|\;\cdots\;|\;\gamma\underline{\textbf{c}},\;|\;\underline{\textbf{c}}),\end{array}$$

$${\tau }_1(\wp (\underline{\text{c}}))$$

$${\text{c}}_{i\ell +j}$$

Full text

1. Introduction

Let R be finite chain ring with invariant $(q,s),$ $\pi:\texttt{R}\rightarrow\mathbb{F}_{q}$ be the natural ring epimorphism, and $\ell$ a positive integer such that $\texttt{gcd}(q,\ell)=1.$ Let $\texttt{R}^{\times}$ be the group of units of $\texttt{R},$ and $\gamma\in\texttt{R}^{\times}.$ An R-linear code $\mathcal{C}$ of length $\ell$ is $\gamma$-constacyclic if $\tau_{\gamma}(\mathcal{C})=\mathcal{C},$ where $\tau_{\gamma}:\texttt{R}^{\ell}\rightarrow\texttt{R}^{\ell},$ is the $\gamma$-constashift operator, defined by $\tau_{\gamma}(\textbf{c}_{0},\textbf{c}_{1},\cdots,\textbf{c}_{\ell-1})=(\gamma\textbf{c}_{\ell-1},\textbf{c}_{0},\cdots,\textbf{c}_{\ell-2}).$ Especially, cyclic and negacyclic linear codes correspond to $\gamma=1_{\texttt{R}}$ and $\gamma=-1_{\texttt{R}},$ respectively (see [4]). The residue code of R-linear code $\mathcal{C}$ is the $\mathbb{F}_{q}$-linear code $\pi(\mathcal{C}):=\left\{(\pi(\textbf{c}_{0}),\pi(\textbf{c}_{1}),\cdots,\pi(\textbf{c}_{\ell-1}))\;:\;(\textbf{c}_{0},\textbf{c}_{1},\cdots,\textbf{c}_{\ell-1})\in\mathcal{C}\right\}.$ The equality $\pi(\tau_{\gamma}(\mathcal{C}))=\tau_{\pi(\gamma)}\left(\pi(\mathcal{C})\right),$ enables to see that the residue code of any $\gamma$-constacyclic R-linear code, is an $\pi(\gamma)$-constacyclic $\mathbb{F}_{q}$-linear code. In the literature [3, 5, 10, 11, 12], the class of $\gamma$-constacyclic R-linear codes, which are studied, have the following property $\gamma\in 1_{\texttt{R}}+\texttt{R}\theta.$

In this paper, on the one hand, we will describe each $\gamma$-constacyclic R-linear code of length $\ell,$ as contraction of a cyclic R-linear code of length $u\ell,$ and on the other hand, we will investigate on the structure of $\gamma$-constacyclic R-linear codes, where $\gamma\in\Gamma(\texttt{R})\setminus\{0_{\texttt{R}}\}.$

The present paper is organized as follows. In Sect. 2, we present results which will be used in the following sections. Sect. 3 studies the subring subcode and trace code of a linear codes over finite chain rings. In Sect.4, the trace-description of cyclic linear codes over finite chain rings is presented. For any $\gamma\in\Gamma(\texttt{R}),$ we proceed to investigate on the structural properties of $\gamma$-constacyclic codes of arbitrary length $\ell,$ in Sect. 5.

2. Background on finite chain rings

Throughout of this section, R is a commutative ring with identity and $\texttt{J}(\texttt{R})$ denoted the Jacobson radical of $\texttt{R},$ and $\texttt{R}^{\times}$ denotes the multiplicative group of units of $\texttt{R}.$ The definitions and results on the finite chain rings are extracted in monographs [6, 8].

Definition 2.1**.**

We say that R is a finite chain ring of invariants $(q,s),$ if:

1. (1)

R* is local principal ideal ring;* 2. (2)

$\texttt{R}/\texttt{J}(\texttt{R})\simeq\mathbb{F}_{q}$* and $\texttt{R}\supsetneq\texttt{R}\theta\supsetneq\cdots\supsetneq\texttt{R}\theta^{s-1}\supsetneq\texttt{R}\theta^{s}=\{0\},$ where $\theta$ is a generator of $\texttt{J}(\texttt{R}).$*

The map $\pi:\texttt{R}\rightarrow\mathbb{F}_{q}$ denotes the canonical projection.

Lemma 1**.**

Let R be a finite chain ring of invariants $(q,s),$ and $\theta$ be a generator of $\texttt{J}(\texttt{R}).$ Then

1. (1)

$\texttt{R}^{\times}=\texttt{R}\setminus\texttt{J}(\texttt{R}),$* and the ideals of R are precisely $\texttt{J}(\texttt{R})^{t}=\texttt{R}\theta^{t},$ where $t\in\{0,1,\cdots,s\};$* 2. (2)

$|\texttt{R}^{\times}|=q^{(s-1)}(q-1)$* and $|\texttt{J}(\texttt{R})^{t}|=q^{s-t},$ for every $t\in\{0,1,\cdots,s\}.$*

Theorem 1**.**

Let R be a finite chain ring of invariants $(q,s),$ and $\theta$ be a generator of $\texttt{J}(\texttt{R}).$ Then

1. (1)

$\texttt{R}^{\times}=\Gamma(\texttt{R})^{*}\cdot(1+\texttt{R}\theta),$* and $\Gamma(\texttt{R})^{*}\simeq\mathbb{F}_{q}\setminus\{0\}$ (as multiplicative group) where $\Gamma(\texttt{R})^{*}:=\{\textbf{b}\in\texttt{R}\,:\,\textbf{b}\neq 0,\,\textbf{b}^{q}=\textbf{b}\};$* 2. (2)

$\Gamma(\texttt{R})^{*}$* is a cyclic subgroup of $\texttt{R}^{\times},$ of order $q-1$ and $|1_{\texttt{R}}+\texttt{R}\theta|=q^{s-1};$* 3. (3)

for every element $\textbf{a}\in\texttt{R},$ there exists a unique $(\textbf{a}_{0},\textbf{a}_{1},\cdots,\textbf{a}_{s-1})\in\Gamma(\texttt{R})^{s},$ such that $\textbf{a}=\textbf{a}_{0}+\textbf{a}_{1}\theta+\cdots+\textbf{a}_{s-1}\theta^{s-1}.$

Definition 2.2**.**

Let R be a finite chain ring of invariants $(q,s),$ and $\theta$ be a generator of $\texttt{J}(\texttt{R}).$ The set $\Gamma(\texttt{R})=\Gamma(\texttt{R})^{*}\cup\{0\}$ is called the Teichm ller set of $\texttt{R}.$

We say that the ring S is an extension of R and we denote it by $\texttt{S}|\texttt{R}$ if R is a subring of S and $1_{\texttt{R}}=1_{\texttt{S}}.$ We denote by $\texttt{rank}_{\texttt{R}}(\texttt{S}),$ the rank of R-module $\texttt{S}.$ We denote by $\texttt{Aut}_{\texttt{R}}(\texttt{S}),$ the group of ring automorphisms of S which fix the elements of $\texttt{R}.$

Definition 2.3**.**

Let R be a finite chain ring of invariants $(q,s).$ We say that the finite chain ring S is the Galois extension of R of degree $m,$ if

1. (1)

$\texttt{S}|\texttt{R}$* is unramified, i.e. $\texttt{J}(\texttt{S})=\texttt{J}(\texttt{R})\texttt{S};$* 2. (2)

$\texttt{S}|\texttt{R}$* is normal, i.e. $\texttt{R}:=\{\textbf{a}\in\texttt{S}\;:\;\varrho(\textbf{a})=\textbf{a}\text{ for all }\varrho\in\texttt{Aut}_{\texttt{R}}(\texttt{S})\}.$*

Proposition 1**.**

Let R be a finite chain ring of invariants $(q,s).$ Let S is the Galois extension of R of degree $m.$ Then

1. (1)

S* is a free R-module of rank $m;$* 2. (2)

$\texttt{Aut}_{\texttt{R}}(\texttt{S})$* is cyclic of order $m;$* 3. (3)

$\texttt{S}=\texttt{R}[\xi]$* where $\xi$ is a generator of $\Gamma(\texttt{S}).$*

Definition 2.4**.**

Let $\texttt{S}|\texttt{R}$ be the Galois extension of finite chain rings of degree $m$ and $\sigma$ be a generator of $\texttt{Aut}_{\texttt{R}}(\texttt{S}).$ The map $\texttt{Tr}_{\texttt{R}}^{\texttt{S}}:=\sum\limits_{i=0}^{m-1}\sigma^{i},$ is called the trace map of the Galois extension $\texttt{S}|\texttt{R}.$

Proposition 2**.**

[6, Chap. XIV]** Let $\texttt{S}|\texttt{T}$ and $\texttt{R}|\texttt{T}$ be Galois extensions of finite chain rings. Then

1. (1)

$\texttt{R}=\left\{\textbf{a}\in\texttt{S}\;:\;\sigma(\textbf{a})=\textbf{a}\;\text{ for all }\sigma\in\texttt{Aut}_{\texttt{R}}(\texttt{S})\right\};$** 2. (2)

the bilinear form $\varphi:(\textbf{a},\textbf{b})\mapsto\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\textbf{a}\textbf{b})$ is nondegenerate; 3. (3)

$\texttt{Tr}_{\texttt{R}}^{\texttt{S}}$* is a generator of S-module $\texttt{Hom}_{\texttt{R}}(\texttt{S},\texttt{R}),$ and $\texttt{Tr}_{\texttt{T}}^{\texttt{R}}\circ\texttt{Tr}_{\texttt{R}}^{\texttt{S}}=\texttt{Tr}_{\texttt{T}}^{\texttt{S}}.$*

3. Linear codes over finite chain rings

Recall that an R-linear code of length $\ell$ is an R-submodule of $\texttt{R}^{\ell}.$ We say that an R-linear code is free if it is a free as R-module.

3.1. Type and rank of a linear code

A matrix $G$ is called a generator matrix for $\mathcal{C}$ if the rows of $G$ span $\mathcal{C}$ and none of them can be written as an R-linear combination of the other rows of $G.$ We say that $G$ is a generator matrix in standard form if

[TABLE]

where $U$ is a suitable permutation matrix. The $s$-tuple $(k_{0},k_{1},\cdots,k_{s-1})$ is called type of $G$ and $\texttt{rank}(G):=k_{0}+k_{1}+\cdots+k_{s-1}$ is the rank of $G.$

Proposition 3**.**

([9, Proposition 3.2, Theorem 3.5]) Each R-linear code $\mathcal{C}$ admits a generator matrix $G$ standard form. Moreover, the type is the same for any generator matrix in standard form for $\mathcal{C}.$

So the type and the rank are the invariants of $\mathcal{C},$ and henceforth we have the following definition.

Definition 3.1**.**

Let $\mathcal{C}$ be an R-linear code.

1. (1)

The type of $\mathcal{C}$ is the type of a generator matrix of $\mathcal{C}$ in standard form. 2. (2)

The rank of $\mathcal{C},$ denoted $\texttt{rank}_{\texttt{R}}(\mathcal{C}),$ is the rank of a generator matrix of $\mathcal{C}$ in standard form.

Obviously, any R-linear code $\mathcal{C}$ of length $\ell$ and of type $(k_{0},k_{1},\cdots,k_{s-1})$ is free if and only if the rank of $\mathcal{C}$ is $k_{0},$ and $k_{1}=k_{2}=\cdots=k_{s-1}=0.$ It defines the scalar product on $\texttt{R}^{\ell}$ by: $\underline{\textbf{a}}\cdot\underline{\textbf{b}}^{\texttt{T}}:=\sum\limits_{i=0}^{\ell-1}\textbf{a}_{i}\textbf{b}_{i},$ where $\underline{\textbf{b}}^{\texttt{T}}$ is the transpose of $\underline{\textbf{b}}.$ Let $\mathcal{C}$ be an R-linear code of length $\ell.$ The dual code of $\mathcal{C},$ denoted $\mathcal{C}^{\perp},$ is an R-linear code of length $\ell,$ define by: $\mathcal{C}^{\perp}:=\left\{\underline{\textbf{a}}\in\texttt{R}^{\ell}\;:\;\underline{\textbf{a}}\cdot\underline{\textbf{b}}^{\texttt{T}}=0\text{ for all }\underline{\textbf{c}}\in\mathcal{C}\right\}.$ A generator matrix of $\mathcal{C}^{\perp},$ is called parity-check matrix of $\mathcal{C}.$

Proposition 4**.**

([9, Theorem 3.10]) Let $\mathcal{C}$ be an R-linear code of length $\ell$ and of type $(k_{0},k_{1},\cdots,k_{s-1}).$ Then

1. (1)

the type of $\mathcal{C}^{\perp}$ is $(\ell-k,k_{s-1},\cdots,k_{1}),$ where $k:=k_{0}+k_{1}+\cdots+k_{s-1}.$ 2. (2)

$|\mathcal{C}|=q^{\sum\limits_{t=0}^{s-1}(s-t)k_{t}},$* where $|\mathcal{C}|$ denotes the number of elements of $\mathcal{C}.$*

3.2. Galois closure of a linear code over a finite chain ring

Let $\mathcal{B}$ be an S-linear codes of length $\ell.$ Then

[TABLE]

is also an S-linear codes of length $\ell.$ We say that the S-linear code $\mathcal{B}$ is called $\sigma$-invariant if $\sigma(\mathcal{B})=\mathcal{B}.$ The subring subcode of $\mathcal{B}$ to $\texttt{R},$ is R-linear code $\texttt{Res}_{\texttt{R}}(\mathcal{B}):=\mathcal{B}\cap\texttt{R}^{\ell},$ and the trace code of $\mathcal{B}$ over $\texttt{R},$ is the R-linear code

[TABLE]

It is clear that $\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\sigma(\mathcal{B}))=\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\mathcal{B}).$ The extension code of an R-linear code $\mathcal{C}$ to $\texttt{S},$ is the S-linear code $\texttt{Ext}_{\texttt{S}}(\mathcal{C}),$ formed by taking all combinations of codewords of $\mathcal{C}.$ The following theorem generalizes Delsarte’s celebrated result (see [13, Ch.7.§8. Theorem 11.]).

Theorem 2**.**

([7, Theorem 3]). Let $\mathcal{B}$ be an S-linear code then $\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\mathcal{B}^{\perp})=\texttt{Res}_{\texttt{R}}(\mathcal{B})^{\perp},$ where $\mathcal{B}^{\perp}$ is the dual to $\mathcal{B}$ with respect to the usual scalar product, and $\texttt{Res}_{\texttt{R}}(\mathcal{B})^{\perp}$ is the dual of $\texttt{Res}_{\texttt{R}}(\mathcal{B})$ in $\texttt{R}^{\ell}.$

Definition 3.2**.**

Let $\mathcal{B}$ be an S-linear code. The $\sigma$-closure of $\mathcal{B},$ is the smallest $\sigma$-invariant S-linear code $\widetilde{\mathcal{B}},$ containing $\mathcal{B}.$

Proposition 5**.**

Let $\mathcal{B}$ be an S-linear code. Then $\widetilde{\mathcal{B}}=\sum\limits_{i=0}^{m-1}\sigma^{i}(\mathcal{B})$ and $\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\mathcal{B})=\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\widetilde{\mathcal{B}}).$

**Proof.**We have $\mathcal{B}\subseteq\widetilde{\mathcal{B}}$ and $\sigma(\widetilde{\mathcal{B}})=\widetilde{\mathcal{B}},$ by Definition 3.2 of $\widetilde{\mathcal{B}}.$ So $\sigma^{i}(\mathcal{B})\subseteq\widetilde{\mathcal{B}},$ for all $i\in\{0,1,\cdots,m-1\}.$ Hence $\sum\limits_{i=0}^{m-1}\sigma^{i}(\mathcal{B})\subseteq\widetilde{\mathcal{B}}.$ Since $\sigma\left(\sum\limits_{i=0}^{m-1}\sigma^{i}(\mathcal{B})\right)=\sum\limits_{i=0}^{m-1}\sigma^{i}(\mathcal{B})$ and $\mathcal{B}\subseteq\sum\limits_{i=0}^{m-1}\sigma^{i}(\mathcal{B}),$ as $\widetilde{\mathcal{B}}$ is the smallest S-linear code containing $\mathcal{B},$ which is $\sigma$-invariant, it follows $\widetilde{\mathcal{B}}\subseteq\sum\limits_{i=0}^{m-1}\sigma^{i}(\mathcal{B}).$ Hence $\widetilde{\mathcal{B}}=\sum\limits_{i=0}^{m-1}\sigma^{i}(\mathcal{B}).$ Thanks to [7, Proposition 1.], $\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\widetilde{\mathcal{B}})=\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\mathcal{B}).$ ∎

The following Theorem summarizes the obtained results in [7].

Theorem 3**.**

Let $\mathcal{B}$ be an S-linear code and $\sigma$ be a generator of $\texttt{Aut}_{\texttt{R}}(\texttt{S})$. Then the following statements are equivalent:

1. (1)

$\mathcal{B}$* is $\sigma$-invariant;* 2. (2)

$\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\mathcal{B})=\texttt{Res}_{\texttt{R}}(\mathcal{B});$** 3. (3)

$\mathcal{B},$* and $\texttt{Res}_{\texttt{R}}(\mathcal{B})$ have the same type.*

**Proof.**Let $\mathcal{B}$ be an S-linear code.

$1.\Leftrightarrow 2.$**: **

Thanks to [7, Theorem 2].

$1.\Leftrightarrow 3.$**: **

Since any R-basis of $\texttt{Res}_{\texttt{R}}(\mathcal{B})$ is also an S-basis of $\texttt{Ext}_{\texttt{S}}\left(\texttt{Res}_{\texttt{R}}(\mathcal{B})\right).$ Thanks to [7, Theorem 1], we deduce that $\mathcal{B}=\texttt{Ext}_{\texttt{S}}\left(\texttt{Tr}_{\texttt{R}}^{\texttt{S}}(\mathcal{B})\right)$ if and only if $\mathcal{B}$ and $\texttt{Res}_{\texttt{R}}(\mathcal{B})$ have the same type.

∎

4. Cyclic linear codes over finite chain rings

Let $\ell$ be a positive integer such that $\texttt{gcd}(q,\ell)=1.$ Then the remainder $q\,(\texttt{mod}\;\ell)$ of $q$ by $\ell,$ belongs to $\mathbb{Z}_{\ell}^{\times},$ the positive integer $m$ denotes the multiplicative order of $q\,(\texttt{mod}\;\ell).$ Let $\Sigma_{\ell}:=\{0,1,\cdots,\ell-1\}$ be the underling set of $\mathbb{Z}_{\ell}.$

4.1. Cyclotomic cosets

Let $u$ be a positive integer. The set of multiples of $u$ in $\mathrm{A}$ is

[TABLE]

The $q$-closure of $\mathrm{A}$ is $\complement_{q}(\mathrm{A}):=\underset{i\in\mathbb{N}}{\cup}q^{i}\mathrm{A}.$

Definition 4.1**.**

Let $z\in\Sigma_{\ell}.$ The $q$-cyclotomic coset modulo $\ell,$ containing $z,$ the Galois closure of $\{z\}.$ We simply write $\complement_{q}(z):=\complement_{q}(\{z\}).$

It denotes by $\Re_{\ell}(q)$ the set of $q$-closure subsets of $\Sigma_{\ell}.$ Obviously, the $q$-cyclotomic cosets modulo $\ell,$ form a partition of $\Sigma_{\ell}.$ Let $\Sigma_{\ell}(q)$ be a set of representatives of each $q$-cyclotomic cosets modulo $\ell.$

Proposition 6**.**

[1, Proposition 5.2]** We have $|\Sigma_{\ell}(q)|=\sum\limits_{d|\ell}\frac{\phi(\ell)}{\texttt{ord}_{\ell}(q)},$ where $\phi(.)$ is the Euler totient function and $\texttt{ord}_{\ell}(q):=\texttt{min}\left\{i\in\mathbb{N}\;:\;q^{i+1}\,\equiv\,1\,(\texttt{mod}\;\ell)\right\}.$

Notation 1**.**

Let $z\in\Sigma_{\ell}$ and $\mathrm{A}$ be a subset of $\Sigma_{\ell}$ and $u\in\mathbb{N},$ with $\texttt{gcd}(u\,,\,q)=1.$

1. (1)

The opposite of $\mathrm{A}$ is $-\mathrm{A}:=\{\ell-z\;:\;z\in\mathrm{A}\}.$ 2. (2)

The complementary of $\mathrm{A}$ is $\overline{\mathrm{A}}:=\left\{z\in\Sigma_{\ell}\;:\;z\not\in\mathrm{A}\right\}.$ 3. (3)

The dual of $\mathrm{A}$ is $\mathrm{A}^{\diamond}:=\overline{-\mathrm{A}}.$

Remark 1**.**

Let $\mathrm{A}$ be a subset of $\Sigma_{\ell}.$ Then $\complement_{q}\left(\overline{\mathrm{A}}\right)=\overline{\complement_{q}(\mathrm{A})}$ and $-\complement_{q}(\mathrm{A})=\complement_{q}(-\mathrm{A}).$ Moreover $(\mathrm{A}^{\diamond})^{\diamond}=\mathrm{A}.$

Example 4.1**.**

We take $\;\ell=20,q=3.$ The $q$-cyclotomic cosets modulo $\ell,$ are: $\complement_{q}(\{0\})=\{0\},\complement_{q}(\{5\})=\{5,15\},\complement_{q}(\{10\})=\{10\},$ and

[TABLE]

So $\Sigma_{\ell}(q)=\{0,1,2,4,5,10,11\}.$ We remark that $\complement_{q}(\{-z\})=\complement_{q}(\{z\}),$ for every $z\in\{0,2,4,5,10\}.$ We set $\;\mathrm{I}:=[0,10].$ We have $\mathrm{A}:=\complement_{q}(\mathrm{I})=\complement_{q}(\{0,1,2,4,5,10\}),$ $-\mathrm{A}=\complement_{q}(\{2,4,5,10,11\}),$ and $\mathrm{A}^{\diamond}:=\complement_{q}(\{1\}).$

4.2. Likewise Reed-Solomon codes over finite chain rings

Let S be the Galois extension of R of degree $m$ and $\xi$ be a generator of $\Gamma(\texttt{S})\backslash\{0\}.$ Let $\mathrm{A}:=\{a_{1},a_{2},\cdots,a_{k}\}$ be a subset of $\Sigma_{\ell}.$ One denotes by $\textbf{P}(\texttt{S}\,;\,\mathrm{A}),$ the free S-module with S-basis $\{X^{a}\,:\,a\in\mathrm{A}\}.$ Since $m$ is the smallest positive integer with $q^{m}\equiv 1\,(\texttt{mod}\,\ell),$ we can write $\eta:=\xi^{\frac{q^{m}-1}{\ell}}$ and the multiplicative order of $\eta$ is $\ell.$ The evaluation

[TABLE]

is an S-modules monomorphism. We see that if $\mathrm{A}:=\{0,1,\cdots,k-1\},$ then for any $\ell^{\texttt{th}}$-primitive root of unity $\eta$ in $\Gamma(\texttt{S}),$ the S-linear code $\textbf{ev}_{\eta}(\textbf{P}(\texttt{S}\,;\,\mathrm{A}))$ is a primitive Reed-Solomon code. For this reason, we define Likewise Reed-Solomon codes which are a family of codes defined over large finite chain rings as follows.

Definition 4.2**.**

Let $\mathrm{A}$ be a subset of $\Sigma_{\ell},$ and S be a finite chain ring such that $|\Gamma(\texttt{S})|\geq\ell.$ Let $\eta\in\Gamma(\texttt{S})$ and the multiplicative order of $\eta$ is $\ell.$ The S-submodule $\textbf{ev}_{\eta}(\textbf{P}(\texttt{S}\,;\,\mathrm{A}))$ is called likewise Reed-Solomon code over $\texttt{S},$ with defining pair $(\eta\;,\;\mathrm{A}).$

We remark that $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}):=\textbf{ev}_{\eta}(\textbf{P}(\texttt{S}\,;\,\mathrm{A}))$ is the free S-linear code with free S-basis $\{\textbf{ev}_{\eta}(X^{a})\;:\;a\in\mathrm{A}\},$ where $\mathrm{A}$ is a subset of $\Sigma_{\ell}.$ We remark that $\textbf{L}_{\eta}(\texttt{S}\,;\,\emptyset)=\{\textbf{0}\},$ $\textbf{L}_{\eta}(\texttt{S}\,;\,\{0\})=\textbf{1}$ and $\textbf{L}_{\eta}(\texttt{S}\,;\,\Sigma_{\ell})=\texttt{S}^{\ell}.$

Proposition 7**.**

Let $\mathrm{A},\mathrm{B}$ be two subsets of $\Sigma_{\ell}.$ Then

1. (1)

$\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})$* is cyclic;* 2. (2)

$\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}\cup\mathrm{B})=\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})+\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{B})$* and $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}\cap\mathrm{B})=\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})\cap\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{B}).$*

Proof.

Consider the codeword $\underline{\textbf{c}}_{a}=\left(1,\eta^{a},\cdots,\eta^{a(\ell-1)}\right).$ Then the shift of $\underline{\textbf{c}}_{a}$ is $\eta^{-a}\underline{\textbf{c}}_{a}.$ Since $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})$ is S-linear, we have $\eta^{-a}\underline{\textbf{c}}_{a}\in\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}).$ Hence $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})$ is cyclic. It is clear that $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}\cup\mathrm{B})\supseteq\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})+\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{B}).$ The set $\{\textbf{ev}_{\eta}(X^{a})\;:\;a\in\mathrm{A}\cup(\mathrm{B}\setminus\mathrm{A})\}$ is a free R-basis of $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}\cup\mathrm{B})$ and $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})+\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{B}).$ Hence, $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}\cup\mathrm{B})=\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})+\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{B}).$ We leave the last equality as an exercise. ∎

Proposition 8**.**

Let $\mathrm{A}$ be a subset of $\Sigma_{\ell}$ and $u$ be a positive integer such that $\texttt{gcd}(\ell,u)=1.$ Then

1. (1)

$\textbf{L}_{\eta^{u}}(\texttt{S}\,;\,\mathrm{A})=\textbf{L}_{\eta}(\texttt{S}\,;\,u\mathrm{A});$** 2. (2)

$\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})^{\perp}=\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}^{\diamond});$** 3. (3)

$\textbf{L}_{\eta}\left(\texttt{S}\,;\,\complement_{q}(\mathrm{A})\right)$* is the $\sigma$-closure of $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}).$*

**Proof.**Assume that $\texttt{gcd}(\ell,u)=1.$ Then $\eta$ and $\eta^{u}$ are $\ell^{\texttt{th}}$-primitive roots of unity. Since $\{\textbf{ev}_{\eta}(X^{a})\;:\;a\in u\mathrm{A}\}$ is a free R-basis of $\textbf{L}_{\eta^{u}}(\texttt{S}\,;\,\mathrm{A}),$ we have $\textbf{L}_{\eta^{u}}(\texttt{S}\,;\,\mathrm{A})=\textbf{L}_{\eta}(\texttt{S}\,;\,u\mathrm{A}).$

A free S-basis of $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}^{\diamond})$ is $\{\underline{\textbf{c}}_{a}\;:\;-a\in\overline{\mathrm{A}}\}$ where $\underline{\textbf{c}}_{a}:=(1,\eta^{-a},\cdots,\eta^{-a(\ell-1)})\in\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}^{\diamond}).$ Then for all $b\in\mathrm{A},$ $\underline{\textbf{c}}_{b}:=(1,\eta^{b},\cdots,\eta^{b(\ell-1)})\in\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}),$ we have $\underline{\textbf{c}}_{b}\underline{\textbf{c}}_{a}^{\texttt{tr}}=\sum\limits_{j=0}^{\ell-1}\eta^{(b-a)j}.$ It is easy to check that $\sum\limits_{j=0}^{\ell-1}\eta^{ij}=0,$ when $i\not\equiv\,0(\texttt{mod}\,\ell).$ Since $0<b-a<\ell,$ we have $\underline{\textbf{c}}_{b}\underline{\textbf{c}}_{a}^{\texttt{tr}}=0.$ So $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}^{\diamond})\subseteq\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})^{\perp}.$ Comparison of cardinality yields $\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})^{\perp}=\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}^{\diamond}).$ Finally, $\sigma(\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}))=\textbf{L}_{\eta}(\texttt{S}\,;\,q\mathrm{A}).$ So by Proposition 5, we have

[TABLE]

Since $\complement_{q}(\mathrm{A})=\overset{m-1}{\underset{i=0}{\bigcup}}q^{i}\mathrm{A},$ we obtain $\widetilde{\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})}=\textbf{L}_{\eta}(\texttt{S}\,;\,\complement_{q}(\mathrm{A})).$ ∎

4.3. Trace representation of free cyclic linear codes

We introduce the map trace-evaluation $\texttt{Tr}_{\texttt{R}}^{\texttt{S}}\circ\textbf{ev}_{\eta}:\textbf{P}_{\eta}(\texttt{S};\mathrm{A})\rightarrow\texttt{R}^{\ell},$ defined by:

[TABLE]

for all $a\in\mathrm{\mathrm{A}}.$ In the sequel, we write: $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}):=\texttt{Tr}_{\texttt{R}}^{\texttt{S}}\left(\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A})\right),$ and $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})$ is a free cyclic R-linear code of length $\ell.$ The immediate proprieties of trace representation of free cyclic linear codes over finite chain ring are given in the following.

Proposition 9**.**

Let $\mathrm{\mathrm{A}},\mathrm{B}$ be two empty subsets of $\Sigma_{\ell}.$ Then

1. (1)

$\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})=\mathrm{\textbf{C}}_{\eta}\left(\texttt{R}\,;\,\complement_{q}(\mathrm{A})\right);$** 2. (2)

$\texttt{rank}_{\texttt{S}}(\textbf{L}_{\eta}(\texttt{S}\,;\,\complement_{q}(\mathrm{A})))=|\complement_{q}(\mathrm{A})|$* and $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})^{\perp}=\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}^{\diamond});$* 3. (3)

$\textbf{C}_{\eta}(\texttt{S}\,;\,\mathrm{A}\cup\mathrm{B})=\textbf{C}_{\eta}(\texttt{S}\,;\,\mathrm{A})+\textbf{C}_{\eta}(\texttt{S}\,;\,\mathrm{B})$* and $\textbf{C}_{\eta}(\texttt{S}\,;\,\mathrm{A}\cap\mathrm{B})=\textbf{C}_{\eta}(\texttt{S}\,;\,\mathrm{A})\cap\textbf{C}_{\eta}(\texttt{S}\,;\,\mathrm{B}).$*

**Proof.**Let $\mathrm{A},\mathrm{B}$ be two subsets of $\Sigma_{\ell}.$

1. (1)

From Proposition 5, $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})=\texttt{Tr}(\textbf{L}_{\eta}(\texttt{S}\,;\,\mathrm{A}))=\texttt{Tr}(\textbf{L}_{\eta}(\texttt{S}\,;\,\complement_{q}(\mathrm{A})))=\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\complement_{q}(\mathrm{A})).$ 2. (2)

Theorem 3(3) yields $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})=\texttt{Tr}(\textbf{L}_{\eta}(\texttt{S}\,;\,\complement_{q}(\mathrm{A})))=\texttt{Res}_{\texttt{R}}(\textbf{L}_{\eta}(\texttt{S}\,;\,\complement_{q}(\mathrm{A}))).$ So

[TABLE]

From Proposition 8, $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})^{\perp}=\mathrm{\textbf{C}}_{\eta}\left(\texttt{R}\,;\,\mathrm{A}^{\diamond}\right).$

∎

The following theorem gives the number of cyclic codes and free cyclic codes over finite chain rings.

Lemma 2**.**

[1, Theorem 5.1]** Let R be a finite chain ring of invariants $(q,s).$ Then the following holds:

1. (1)

the number of cyclic R-linear codes of length $\ell,$ is equal to $(s+1)^{|\Sigma_{\ell}(q)|},$ 2. (2)

the number of free cyclic R-linear codes of length $\ell,$ is equal to $2^{|\Sigma_{\ell}(q)|}.$

Lemma 3**.**

Let R be a finite chain ring of invariants $(q,s)$ and S be the Galois extension of R of degree $m.$ Let $z\in\Sigma_{\ell}.$ Set $\texttt{S}=\texttt{R}[\xi],$ $m_{z}:=|\complement_{q}(z)|,$ $\eta:=\xi^{\frac{q^{m}-1}{\ell}}.$ and $\zeta:=\eta^{-z}.$ Then the map

[TABLE]

is an R-module isomorphism. Further $\texttt{R}[\xi^{m_{z}}]$ is the Galois extension of R of degree $m_{z}$ and $\psi_{z}\circ t_{\zeta}=\tau_{1}\circ\psi_{z},$ where $t_{\zeta}(\textbf{a})=\textbf{a}\zeta,$ for all $\textbf{a}\in\texttt{R}[\eta].$

Proof.

It is clear that $a\in\texttt{Ker}(\psi_{z})$ if and only if $a\in\texttt{R}[\xi^{m_{z}}]^{\perp_{\texttt{Tr}}}\cap\texttt{R}[\xi^{m_{z}}],$ where duality $\perp_{\texttt{Tr}}$ is with respect to trace form. As the trace bilinear form is nondegenerate, we have $\texttt{S}=\texttt{R}[\xi^{m_{z}}]^{\perp_{\texttt{Tr}}}\oplus\texttt{R}[\xi^{m_{z}}]$ and $\texttt{Ker}(\psi_{z})=\{0\}.$ Hence $\psi_{z}$ is an R-module monomorphism. We remark that, $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\})$ is cyclic, if and only if $\psi_{z}\circ t_{\zeta}=\tau_{1}\circ\psi_{z},$ for all $a\in\texttt{R}[\eta].$ Finally, we have $\texttt{S}=\texttt{R}[\xi],$ so $\texttt{R}[\xi^{m_{z}}]$ is the Galois extension of R of degree $m_{z}.$ Hence, $\psi_{z}$ is an R-module isomorphism. ∎

Definition 4.3**.**

A non trivial cyclic R-linear code $\mathcal{C}$ is said to be irreducible, if for all R-linear cyclic subcodes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ of $\mathcal{C},$ such that, $\mathcal{C}=\mathcal{C}_{1}\oplus\mathcal{C}_{2},$ implies $\mathcal{C}_{1}=\{\textbf{0}\}$ or $\mathcal{C}_{2}=\{\textbf{0}\}.$

Proposition 10**.**

The irreducible cyclic R-linear codes are precisely $\theta^{t}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\})$s, where $t\in\{0,1,\cdots,s-1\}$ and $z\in\Sigma_{\ell}(q).$

**Proof.**By Lemma 3, the cyclic R-linear code $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\}))$ and all the R-linear cyclic subcodes are irreducible. Let $\mathcal{C}$ be an irreducible cyclic R-linear code. Then the R-linear code $\texttt{Quot}_{s-1}(\mathcal{C}):=\left\{\textbf{c}\in\texttt{R}^{\ell}\;:\;\theta^{s-1}\textbf{c}\in\mathcal{C}\right\}$ is cyclic and free, and so $\texttt{Quot}_{s-1}(\mathcal{C})=\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})$ for some $\mathrm{A}\subset\Sigma_{\ell}(q)$ and $\mathrm{A}\neq\emptyset.$ Assume that $|\mathrm{A}|>1.$ Then $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})=\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{1})\oplus\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{2})$ where $\mathrm{A}_{1}\cap\mathrm{A}_{2}=\emptyset,$ $\mathrm{A}_{1}\neq\emptyset$ and $\mathrm{A}_{2}\neq\emptyset.$ We have $\mathcal{C}\cap\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{1})\neq\{\textbf{0}\}$ and $\mathcal{C}\cap\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{2})\neq\{\textbf{0}\}.$ Therefore $\mathcal{C}=(\mathcal{C}\cap\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{1}))\oplus(\mathcal{C}\cap\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{2})).$ It is impossible, because $\mathcal{C}$ be an irreducible. So $|\mathrm{A}|=1.$ Now, $\mathcal{C}\subseteq\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\}),$ it follows that $\mathcal{C}=\theta^{t}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\}),$ for some $t\in\{0,1,\cdots,s-1\}.$ ∎

We set $\Sigma_{\ell}(q)$ a set of representatives of each $q$-cyclotomic cosets modulo $\ell.$ An $(q,s)$-cyclotomic partition modulo $\ell,$ is the $(s+1)$-tuple $(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})$ with the property $\mathrm{A}_{t}=\complement_{q}\left(\lambda^{-1}(\{t\})\right),$ where $\lambda:\Sigma_{\ell}(q)\rightarrow\{0,1,\cdots,s\}$ is a map. Denoted by

[TABLE]

the set of $(q,s)$-cyclotomic partitions modulo $\ell,$ and $\texttt{Cy}(\texttt{R},\ell)$ the set of cyclic R-linear codes of length $\ell.$ We have $|\Re_{\ell}(q,s)|=(s+1)^{|\Sigma_{\ell}(q)|}.$

Example 4.2**.**

We take $\;\ell=20,q=3$ and $s=2.$ Then $|\Sigma_{\ell}(q)|=13$ and $|\Re_{\ell}(q,s)|=3^{7}.$ An $(q,s)$-cyclotomic partition modulo $\ell,$ is $\underline{\mathrm{A}}:=\left(\complement_{q}(\{0,1,2\}),\complement_{q}(\{5,11\}),\complement_{q}(\{4,10\})\right).$

Theorem 4**.**

Any cyclic R-linear code $\mathcal{C}$ there exists a unique $\mathrm{A}:=(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})\in\Re_{\ell}(q,s)$ such that $\mathcal{C}=\mathrm{\textbf{C}}_{\texttt{R}}(\mathrm{A})$ and $\mathrm{\textbf{C}}_{\texttt{R}}(\mathrm{A})=\overset{s-1}{\underset{t=0}{\bigoplus}}\theta^{t}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{t}).$ Moreover, the type of $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})$ is

[TABLE]

for some $\underline{\mathrm{A}}:=(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})\in\Re_{\ell}(q,s).$

**Proof.**Let $\mathcal{C}$ be an cyclic R-linear code of length $\ell.$ From Proposition 9, we have $\texttt{R}^{\ell}=\underset{z\in\Sigma_{\ell}(q)}{\bigoplus}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\})$ and $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\})$’s are free irreducible cyclic R-linear codes. Therefore $\mathcal{C}=\underset{z\in\Sigma_{\ell}(q)}{\bigoplus}\mathcal{C}_{z},$ where $\mathcal{C}_{z}=\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\})\cap\mathcal{C}.$ From Proposition 10, $\mathcal{C}_{z}=\theta^{t_{z}}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\}),$ where $t_{z}\in\{0,1,\cdots,s\}.$ Hence

[TABLE]

where $\mathrm{A}_{t}=\{z\in\Sigma_{\ell}\;:\;t_{z}=t\}.$ Since $|\Re_{\ell}(q,s)|=(s+1)^{|\Sigma_{\ell}(q)|},$ by Theorem 2, the uniqueness of $\underline{\mathrm{A}}:=(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})\in\Re_{\ell}(q,s)$ such that $\mathcal{C}=\mathrm{\textbf{C}}_{\texttt{R}}(\mathrm{A})$ is guaranteed.

Moreover, for every $t\in\{0,1,\cdots,s-1\},$ the cyclic R-linear code $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{t})$ is free and $\texttt{rank}_{\texttt{R}}(\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{t}))=|\complement_{q}(\mathrm{A}_{t})|.$ Since the direct sum $\overset{s-1}{\underset{t=0}{\bigoplus}}\theta^{t}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{t})$ gives the type of $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}}),$ the type of $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})$ is $(k_{0},k_{1},\cdots,k_{s-1}),$ where $k_{t}:=|\complement_{q}(\mathrm{A}_{t})|,$ for every $t\in\{0,1,\cdots,s-1\}.$ ∎

Proposition 11**.**

Let $\underline{\mathrm{A}}:=(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})\in\Re_{\ell}(q,s)$ and $t\in\{0,1,\cdots,s-1\}.$ Then $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})^{\perp}=\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}}^{\widetilde{\diamond}}),$ where $\underline{\mathrm{A}}^{\widetilde{\diamond}}:=(-\mathrm{A}_{s},-\mathrm{A}_{s-1},\cdots,-\mathrm{A}_{1},-\mathrm{A}_{0}).$

**Proof.**Let $\underline{\mathrm{A}}:=(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})\in\Re_{\ell}(q,s).$ We have $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})^{\perp}\supseteq\bigcap_{u=0}^{s-1}\left(\theta^{s-u}\texttt{R}^{\ell}+\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{u}^{\diamond})\right)$ and $\theta^{s-t}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,-\mathrm{A}_{t})\subseteq\bigcap_{u=0}^{s-1}\left(\theta^{s-u}\texttt{R}^{\ell}+\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}_{u}^{\diamond})\right),$ for every $t\in\{1,2,\cdots,s\}.$ It follows that $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}}^{\widetilde{\diamond}})\subseteq\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})^{\perp}.$ From Proposition 4 and Theorem 4, $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}}^{\widetilde{\diamond}})$ and $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})^{\perp}$ have the same type, we have $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})^{\perp}=\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}}^{\widetilde{\diamond}}).$ ∎

5. Constacyclic linear codes over a finite chain ring

Let $\gamma\in\texttt{R}^{\times}$ and the multiplicative order of $\gamma$ is $u.$ We study the structure of contractions of cyclic R-linear codes of length $u\ell.$ In the section, the usage of the map

[TABLE]

will be necessary.

Definition 5.1**.**

Let $\mathcal{C}$ be an R-linear code of length $u\ell$ and $\mathcal{C}:=\wp(\mathcal{K}).$

1. (1)

The R-linear code $\mathcal{K}$ is called the contraction of a linear code of $\mathcal{C}.$ 2. (2)

The R-linear code $\mathcal{C}$ is called the cyclic concatenation of $\mathcal{K}.$

The contraction of a class of linear cyclic codes over finite fields have been investigated in [2]. Our contribution is the generalization of this theory of contraction of cyclic codes to finite chain rings.

Lemma 4**.**

Let $\gamma\in\texttt{R}^{\times}$ and $u$ the multiplicative order of $\gamma.$ Then the map $\wp$ is an R-module monomorphism. Moreover $\wp\circ\tau_{\gamma}=\tau_{1}\circ\wp.$

**Proof.**It is clear that $\wp$ is an R-module monomorphism. Let $\underline{\textbf{c}}:=(\textbf{c}_{0},\cdots,\textbf{c}_{\ell-1})\in\texttt{R}^{\ell}$ and $\tau_{1}$ the cyclic shift on $\{0,1,\cdots,u\ell-1\}.$ We have:

[TABLE]

Hence $\wp\circ\tau_{\gamma}=\tau_{1}\circ\wp.$ ∎

Corollary 1**.**

Let $\gamma\in\texttt{R}^{\times}$ and $u$ the multiplicative order of $\gamma$ and $\mathcal{K}$ be an R-linear code of length $\ell.$ Then $\mathcal{K}$is $\gamma$-constacyclic if and only if $\wp(\mathcal{K})$ is cyclic R-linear code of length $u\ell.$ Moreover, $\mathcal{K}$ and $\wp(\mathcal{K})$ have the same type.

**Proof.**The map $\wp$ is an R-module monomorphism. So $\wp(\mathcal{K})$ is R-linear code of length $u\ell$ and $\mathcal{K},$ $\wp(\mathcal{K})$ have the same type. From Lemma 4, we have $\wp\circ\tau_{\gamma}=\tau_{1}\circ\wp,$ and so $\wp(\mathcal{K})$ is cyclic. ∎

This show how to construct a cyclic R-linear code from a constacyclic R-linear code. Now we want to construct a constacyclic R-linear code from a cyclic R-linear code. Let $\mathrm{A}$ be a subset of $\{0,1,\cdots,u\ell-1\}.$ One denotes $\mathrm{A}\,(\texttt{mod}\,u):=\{a\,(\texttt{mod}\,u)\;:\;a\in\mathrm{A}\}.$

Theorem 5**.**

Let $u,\ell\in\mathbb{N}$ such that $\texttt{gcd}(u\ell,q)=1.$ Let $\mathrm{A}$ be a subset of $\{0,1,\cdots,u\ell-1\}$ and $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})$ be a cyclic R-linear code of length $u\ell.$ Then $\complement_{q}(\mathrm{A})\,(\texttt{mod}\,u)=\{\omega\},$ if and only if $\mathcal{K}:=\wp^{-1}(\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A}))$ is an $\gamma$-constacyclic R-linear code of length $\ell,$ where $\gamma=\xi^{-\frac{\omega(q^{m}-1)}{u}\,\texttt{mod}\,u}.$ Moreover $\mathcal{K}^{\perp}=\wp^{-1}\left(\mathrm{\textbf{C}}_{\eta}\left(\texttt{R}\,;\,\mathrm{A}^{\star u}\right)\right),$ where $\complement_{q}(\mathrm{A})\,(\texttt{mod}\,u)=\{\omega\},$ and $\mathrm{A}^{\star u}:=\left\{a\in\mathrm{A}^{\diamond}\;:\;a\equiv-\omega(\texttt{mod}\,u)\right\},$ is an $\gamma^{-1}$-constacyclic R-linear code of length $\ell.$

**Proof.**Let $m$ be the positive integer such that $q^{m}\equiv 1\,(\,\texttt{mod}\,u\ell)$ and $q^{m}\not\equiv 1\,(\,\texttt{mod}\,u\ell).$ Let $\texttt{S}:=\texttt{R}[\xi]$ be a Galois extension of R of degree $m.$ We set $\beta:=\xi^{\frac{q^{m}-1}{u}\,\texttt{mod}\,u},$ $w:=\frac{q^{m}-1}{u\ell}$ and $\eta:=\xi^{w}.$ Let $\mathrm{Z}:=\complement_{q}(\mathrm{A}),$ where $\mathrm{A}$ is a subset of $\Sigma_{\ell}.$ Then $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{Z})=\oplus_{z\in\mathrm{Z}}\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\}).$ It is enough to show that $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\})\subseteq\wp(\texttt{R}^{\ell}),$ for all $z\in\mathrm{Z}.$ Let $z\in\mathrm{Z},$ we set $m_{z}:=|\complement_{q}(z)|$ and $\zeta:=\eta^{m_{z}}.$ From Lemma  3, $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\})=\psi_{z}(\texttt{R}[\xi^{m_{z}}])=\texttt{Tr}_{\texttt{R}}^{\texttt{S}}\left(\emph{ev}_{\eta}(\texttt{R}[\xi^{m_{z}}]X^{z})\right).$ Thus for all $\underline{\textbf{c}}:=(\textbf{c}_{0},\cdots,\textbf{c}_{u\ell-1})\in\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\{z\}),$ From Lemma 3, exist a unique $\textbf{a}\in\texttt{R}[\xi^{m_{z}}]$ and such that $\underline{\textbf{c}}=\texttt{Tr}_{\texttt{R}}^{\texttt{S}}\left(\emph{ev}_{\eta}(\textbf{a}X^{z})\right).$ Since $\texttt{R}[\xi^{m_{z}}]$ is the Galois extension of R of degree $m_{z},$ then there exist a unique $(\textbf{a}_{0},\textbf{a}_{1},\cdots,\textbf{a}_{m_{z}-1})$ such that $\textbf{a}:=\sum\limits_{h=0}^{m_{z}-1}\textbf{a}_{h}\xi^{hm_{z}}\in\texttt{R}[\xi^{m_{z}}]$ and $\textbf{c}_{t}:=\sum\limits_{h=0}^{m_{z}-1}\textbf{a}_{h}\texttt{Tr}_{\texttt{R}}^{\texttt{S}}\left(\xi^{hm_{z}+wtz\,\texttt{mod}\,\ell}\right),$ for all $t\in\Sigma_{u\ell}.$ From the euclidian division of $t\in\{0,1,\cdots,u\ell-1\},$ by $\ell,$ there exists $(i,j)\in\Sigma_{u}\times\Sigma_{\ell},$ such that $t=i\ell+j.$ We have:

[TABLE]

Thus $\textbf{c}:=(\cdots\;|\;\gamma^{i}\textbf{x}_{0},\cdots,\gamma^{i}\textbf{x}_{\ell-1}\;|\;\gamma^{i-1}\textbf{x}_{0},\cdots,\gamma^{i-1}\textbf{x}_{\ell-1}\;|\;\cdots)$ and $\gamma:=\beta^{-\omega}.$ Hence $\mathrm{\textbf{C}}_{\eta}(\texttt{R}\,;\,\mathrm{A})\subseteq\wp(\texttt{R}^{\ell}).$ As $\wp\circ\tau_{\gamma}=\tau_{1}\circ\wp,$ it follows that $\mathcal{K}$ is an $\gamma$-constacyclic R-linear code of length $\ell.$ For sufficiency, it is enough to note that the above proof is reversible.

As $\mathcal{K}^{\perp}$ is an $\gamma^{-1}$-constacyclic free R-linear code of rank $\ell-|\mathrm{A}|,$ the cyclic R-linear code which yields $\mathcal{K}^{\perp},$ by contraction must have the definition set $\mathrm{B}$ of size $|\mathrm{A}|.$ ∎

Example 5.1**.**

Let R be a finite chain ring of invariants $(q,s)$ where $q=3.$ We take $\;\ell=28,u=2.$ We set $\mathrm{A}_{1}:=\complement_{q}(\{1,7\}),$ $\mathrm{A}_{2}:=\complement_{q}(\{1,5,7\}),$ and $\mathrm{A}_{3}:=\complement_{q}(\{1,5,7,11\}).$ We have $\complement_{q}(\mathrm{A}_{i})\,(\,\texttt{mod}\;2)=\{1\}.$ So we can set $\mathcal{K}_{i}:=\wp^{-1}\left(\mathrm{\textbf{C}}_{\eta}\left(\texttt{R}\,;\,\mathrm{A}_{i}\right)\right),$ where $i\in\{1,2,3\}.$ Since $\mathrm{A}_{1}^{\star 2}=\mathrm{A}_{3}$ and $\mathrm{A}_{2}^{\star 2}=\mathrm{A}_{2},$ we have $\mathcal{K}_{3}=\mathcal{K}_{1}^{\perp}$ and $\mathcal{K}_{2}$ is self-dual.

The Hamming weight of an R-linear code $\mathcal{C}$ of length $\ell,$ is defined as: $\texttt{wt}(\mathcal{C}):=\texttt{min}\left\{\texttt{wt}(\underline{\textbf{c}})\,:\,\textbf{c}\in\mathcal{C}\setminus\{\textbf{0}\}\right\},$ where $\texttt{wt}(\underline{\textbf{c}}):=|\{j\in\Sigma_{\ell}\;:\;\textbf{c}_{j}\neq 0\}|.$

Corollary 2**.**

Let $u,\ell\in\mathbb{N}$ such that $\texttt{gcd}(u\ell,q)=1.$ Let $\underline{\mathrm{A}}:=(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})\in\Re_{u\ell}(q,s)$ and $\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})$ be a cyclic R-linear code of length $u\ell$ such that $\overset{s-1}{\underset{t=0}{\bigcup}}\mathrm{A}_{t}\,(\,\texttt{mod}\,u)=\{\omega\}.$ Set $\mathcal{K}:=\wp^{-1}\left(\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}})\right).$ Then

1. (1)

$\mathcal{K}$* is an $\gamma$-constacyclic R-linear code of length $\ell,$ where $\gamma=\xi^{-\frac{\omega(q^{m}-1)}{u}\,\texttt{mod}\,u};$* 2. (2)

$\texttt{wt}(\underline{\textbf{c}})=u\cdot\texttt{wt}(\wp^{-1}(\underline{\textbf{c}})),$* for every $\underline{\textbf{c}}\in\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}});$* 3. (3)

$\mathcal{K}^{\perp}=\wp^{-1}(\mathrm{\textbf{C}}_{\texttt{R}}(\underline{\mathrm{A}}^{\star u})),$* where $\underline{\mathrm{A}}^{\star u}:=(-\mathrm{A}_{s}^{\star u},-\mathrm{A}_{s-1},\cdots,-\mathrm{A}_{1},-\mathrm{A}_{0}^{\triangleleft u})$ with*

• •

$\mathrm{A}_{s}^{\star u}:=\left\{a\in\mathrm{A}_{s}\;:\;a\equiv-\omega(\,\texttt{mod}\,u)\right\},$**

• •

$\mathrm{A}_{0}^{\triangleleft u}:=\mathrm{A}_{0}\cup(\mathrm{A}_{s}\setminus\mathrm{A}_{s}^{\star u}).$**

Example 5.2**.**

Let R be a finite chain ring of invariants $(q,s)$ where $\texttt{J}(\texttt{R})=\texttt{R}\theta,$ $q=3$ and $s=2.$ We take $\;\ell=10,u=2.$ We set $\underline{\mathrm{A}}:=(\mathrm{A}_{0},\mathrm{A}_{1},\mathrm{A}_{2}),$ where $\mathrm{A}_{0}:=\complement_{q}(\{1\}),$ $\mathrm{A}_{1}:=\complement_{q}(\{5\}),$ and $\mathrm{A}_{2}:=\complement_{q}(\{0,2,4,10,11\}).$ We have $\complement_{q}(\mathrm{A}_{0})\,(\,\texttt{mod}\;2)=\complement_{q}(\mathrm{A}_{1})\,(\,\texttt{mod}\;2)=\{1\}.$ So the contraction of the cyclic R-linear code $\textbf{C}_{\texttt{R}}(\underline{\mathrm{A}})$ of length $20,$ is the self-dual negacyclic R-linear code $\mathcal{K}:=\wp^{-1}\left(\mathrm{\textbf{C}}_{\eta}\left(\texttt{R}\,;\,\mathrm{A}_{0}\right)\right)\oplus\theta\wp^{-1}\left(\mathrm{\textbf{C}}_{\eta}\left(\texttt{R}\,;\,\mathrm{A}_{1}\right)\right),$ of length $10.$

6. Conclusion

We have seen that in the case $\texttt{gcd}(\ell,|\texttt{R}|)=1,$ and $\gamma\in\Gamma(\texttt{R})^{*},$ the class of $\gamma$-constacyclic R-linear codes of length $\ell,$ is the same as the class of contractions of cyclic R-linear codes $\mathrm{\textbf{C}}_{\texttt{R}}(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})$ of length $u\ell,$ where $u$ is the multiplicative order of $\gamma,$ and each cyclic R-linear code $\mathrm{\textbf{C}}_{\texttt{R}}(\mathrm{A}_{0},\mathrm{A}_{1},\cdots,\mathrm{A}_{s})$ of this class, satisfies: $\overset{s-1}{\underset{t=0}{\bigcup}}\mathrm{A}_{t}\,(\,\texttt{mod}\,u)$ is a singleton.