© Wuhan University 2025

0 Introduction

Random linear network coding, first introduced in Ref. [1], is a strong tool for effective data transmission over noisy and lossy networks. It was proved in Ref. [1] that the information rate of a network can be improved by using coding at the nodes of the network, instead of simply routing the received inputs. An algebraic approach to random network coding was provided by Koetter and Kschischang in Ref. [2]. They proposed transmitting information by means of the subspaces of finite fields ${\mathbb{F}}_q^n$ and defined subspace codes as a class of codes well suited for error correction. In the case that all the codewords in a subspace code have the same dimension, the subspace code is said to be a constant dimension subspace code. The theory of constant dimension subspace code has received a lot of attention in recent years (see Refs. [3-9]). As we know, the approach of constructing good constant dimension subspace codes is an interesting research field. In Ref. [9], Trautmann et al introduced the concept of orbit codes as subspace codes obtained from the action of subgroups of the general linear group $\mathrm{G}\mathrm{L}(n,q)$ on the set of subspaces of ${\mathbb{F}}_q^n$. When the acting group is cyclic, the code is called a cyclic orbit code. Because of the simplicity of their algebraic structure and the existence of efficient encoding/decoding algorithms, this family of codes has attracted great interest. Gluesing-Luerssen and Lehmann^[3] presented a detailed study of cyclic orbit codes based on the stabilizer subfield. Later, Gluesing-Luerssen et al^[4] investigated the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of ${\mathbb{F}}_q^n$ on an ${\mathbb{F}}_q$-subspace $\mathcal{u}$ of ${\mathbb{F}}_q^n$. Ref. [10] gave a systematic construction of subspace codes using subspace polynomials. By using Ben-Sasson’s idea, Chen et al^[6] also provided some constructions of cyclic subspace codes. Roth et al^[11] and Zhang et al^[12] generalized and improved their result, so that one can obtain larger codes for fixed parameters and increase the density of some possible parameters.

Linear codes over finite rings have played a very important role in the theory of error correcting codes and practice (see Refs. [13-19]). On the one hand, by means of linear codes over finite rings, one can obtain good linear codes over finite fields (see Refs. [20-22]). On the other hand, new quantum codes and entanglement-assisted quantum codes can be obtained from linear codes over finite rings (see Refs. [23-27]).

Inspired by these works, in this paper, we first generalize subspace codes over finite fields to submodule codes over finite chain rings, and generalize constant dimension codes over finite fields to constant rank codes over finite chain rings. Under suitable conditions, we give a characterization of the constant rank codes over finite chain rings. We give a sufficient condition for which an orbit submodule code over a finite chain ring is a constant rank code.

This paper is organized as follows. In Section 1, we recall the necessary background materials of linear codes over finite chain rings. In Section 2, we first generalize the constant dimension codes over finite fields to constant rank codes over finite chain rings. Then, we give a relationship between constant rank codes over finite chain rings and constant dimension codes over the residue fields. By means of this relationship, we obtain a method to construct constant rank codes over finite chain rings. In Section 3, we collect concepts of orbit codes over finite fields, which are generalized to the orbit submodule codes over finite chain rings. Then we study orbit submodule constant rank codes over finite chain rings. We give two new examples of the open problem proposed by Gluesing-Luerssen et al in Ref. [4] (see Examples 1 and 2). In Section 4, we define a Gray map $\Phi$ from ${\left({\mathbb{F}}_q+\gamma {\mathbb{F}}_q\right)}^n$ to ${\mathbb{F}}_q^{\mathrm{2}n}$, and we give a method for constructing an optimum distance constant dimension code over ${\mathbb{F}}_q$ by using cyclic codes over ${\mathbb{F}}_q+\gamma {\mathbb{F}}_q$. Finally, a brief summary of this work is described in Section 5.

1 Linear Codes over Chain Rings

Throughout this paper $\mathit{ℛ}$ will denote a finite chain ring. In this section, we recall some basic concepts and results of linear codes over $\mathit{ℛ}$, necessary for the development of this work. For more details, we refer to Refs. [16, 19, 28-30].

It is well known that $\mathit{ℛ}$ has the unique maximal ideal, denoted by $\mathit{m}$. Let $\gamma$ be a generator of the unique maximal ideal $\mathit{m}$, i.e., $\mathit{m}=〈\gamma 〉$. Its chain of ideals is

$\mathit{ℛ}=〈{\gamma }^{\mathrm{0}}〉\supset 〈{\gamma }^{\mathrm{1}}〉\supset \cdots \supset 〈{\gamma }^{t-\mathrm{1}}〉\supset 〈{\gamma }^t〉=\left\{\mathrm{0}\right\}.$

The integer $t$ is called the nilpotency index of $\mathit{m}$. Let ${\mathbb{F}}_q=\mathit{ℛ}/\mathit{m}$ be the residue field with characteristic $p$, where $q=p^{\mathrm{e}}$ and $p$ is a prime number. There is a natural homomorphism from $\mathit{ℛ}$ onto ${\mathbb{F}}_q=\mathit{ℛ}/\mathit{m}$, i.e.,

$\stackrel{}{¯}:\mathit{ℛ}\to {\mathbb{F}}_q=\mathit{ℛ}/\mathit{m}$, $r\mapsto r+\mathit{m}=\overline{r}$, for any $r\in \mathit{ℛ}$.

This natural homomorphism from $\mathit{ℛ}$ onto ${\mathbb{F}}_q=\mathit{ℛ}/\mathit{m}$ can be extended naturally to a homomorphism from ${\mathit{ℛ}}^n$ onto ${\mathbb{F}}_q^n$. For an element $\mathit{c}\in {\mathit{ℛ}}^n$, let $\overline{\mathit{c}}$ be its image under this homomorphism.

Let ${\mathit{ℛ}}^{\mathrm{*}}$ denote the group of units of $\mathit{ℛ}$. ${\mathit{ℛ}}^{\mathrm{*}}$ is just the set of non-nilpotent elements of $\mathit{ℛ}$ i.e., ${\mathit{ℛ}}^{\mathrm{*}}=\mathit{ℛ}-\mathit{m}$. The subgroup $\mathrm{1}+\mathit{m}$ of ${\mathit{ℛ}}^{\mathrm{*}}$ is a $p$-group.

It is well known that the group of units ${\mathit{ℛ}}^{\mathrm{*}}$ of $\mathit{ℛ}$ contains a unique cyclic subgroup $T^{\mathrm{*}}$ of order $q-\mathrm{1}$. $T=T^{\mathrm{*}}\bigcup \{\mathrm{0}\}$ is called the Teichmüller set of $\mathit{ℛ}$ and forms a system of coset representatives of ${\mathbb{F}}_q=\mathit{ℛ}/\mathit{m}$. More precisely, $\mathit{ℛ}$ contains a unit element $\zeta$ with multiplicative order $q-\mathrm{1}$ such that $T=\{\mathrm{0},\mathrm{1},\zeta ,{\zeta }^{\mathrm{2}},\dots ,{\zeta }^{q-\mathrm{2}}\}$. We call $\zeta$ the generator of $T$. Since the set $T$ modulo $\gamma$ equals ${\mathbb{F}}_q$, we do not make distinction between $T$ and ${\mathbb{F}}_q$. Every element $r\in \mathit{ℛ}$ can be written as $r={\displaystyle \sum _{i=\mathrm{0}}^{t-\mathrm{1}}}{r}_i{\gamma }^i$, where $r_{\mathrm{0}},r_{\mathrm{1}},\dots ,r_{t-\mathrm{1}}\in T$ (see Refs. [7,18]).

Lemma 1   Let notations be as above. We have

${\mathit{ℛ}}^{\mathrm{*}}=T^{\mathrm{*}}\cdot (\mathrm{1}+\mathit{m})\cong T^{\mathrm{*}}\times \left(\mathrm{1}+\mathit{m}\right).$

A nonempty subset $C\subseteq {\mathit{ℛ}}^n$ is called a linear code of length $n$ over $\mathit{ℛ}$ if it is an $\mathit{ℛ}$-submodule of ${\mathit{ℛ}}^n$. All codes are assumed to be linear. We say that a linear code $C$ over $\mathit{ℛ}$ is free if $C$ is isomorphic as a module to ${\mathit{ℛ}}^{\mu }$ for some positive lnteger $\mu$, denoted by $C\cong {\mathit{ℛ}}^{\mu }$.

For two vectors $\mathit{a}=(a_{\mathrm{1}},a_{\mathrm{2}},\cdots ,a_n)$ and $\mathit{b}=(b_{\mathrm{1}},b_{\mathrm{2}},\cdots ,b_n)$ in ${\mathit{ℛ}}^n$, we define the Euclidean inner product as $\left[\mathit{a},\mathit{b}\right]$ to be $\left[\mathit{a},\mathit{b}\right]={\displaystyle \sum _{i=\mathrm{1}}^n}{a}_i{b}_i$.

Let $C$ be a linear code over ${\mathit{ℛ}}^n$. We define the Euclidean dual code of $C$ as

$C^{\perp }=\{\mathit{a}\in {\mathit{ℛ}}^n|\left[\mathit{a},\mathit{b}\right]=\mathrm{0}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\mathit{b}\in C\}.$

The following lemma is well-known in Ref. [30].

Lemma 2   Let $C$ be a linear code of length $n$ over $\mathit{ℛ}$ (or an $\mathit{ℛ}$-submodule of ${\mathit{ℛ}}^n$). Then

$\left|C\right|\left|C^{\perp }\right|={\left|\mathit{ℛ}\right|}^n.$

One of the most important tools in coding theory is finding a generator matrix for a code. In general, we want not only a matrix that generates code by rows, but also a matrix that generates code by a minimum number of rows. To describe the generator matrix for a code over $\mathit{ℛ}$, we introduce the following two definitions and lemmas which come from Refs. [16, 28].

Definition 1   Let ${\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_k$ be nonzero vectors in ${\mathit{ℛ}}^n$. Then ${\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_k$ are $\mathit{ℛ}$-independent if ${\displaystyle \sum _{j=\mathrm{1}}^k}{\delta }_i{\mathit{w}}_i=\mathrm{0}$ implies that ${\delta }_j{\mathit{w}}_j=\mathrm{0}$ for all $j$, where ${\delta }_j\in \mathrm{ℛ}$.

Following Definition 1, we can easily get the following lemma.

Lemma 3   If the nonzero vectors ${\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_s$ in ${\mathit{ℛ}}^n$ are $\mathit{ℛ}$-independent and ${\displaystyle \sum _{j=\mathrm{1}}^s}{\delta }_i{\mathit{w}}_i=\mathrm{0}$, then ${\delta }_j\in 〈\gamma 〉$ for all $j$.

Let ${\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_s$ be vectors in ${\mathit{ℛ}}^n$. As usual, we denote the set of all linear combinations of ${\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_s$ by $〈{\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_s〉$.

Lemma 4   If the nonzero vectors ${\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_s$ in ${\mathit{ℛ}}^n$ are $\mathit{ℛ}$-independent, then none of the vectors ${\mathit{w}}_{\mathrm{1}},\dots ,{\mathit{w}}_s$ is a linear combination of the other vectors.

With the help of the above definition and two lemmas, we give a definition of a generator matrix for a code over $\mathit{ℛ}$.

Definition 2   Let $C\ne \left\{\mathrm{0}\right\}$ be a linear code over $\mathit{ℛ}$ (or an $\mathit{ℛ}$-submodule of ${\mathit{ℛ}}^n$). The nonzero codewords ${\mathit{c}}_{\mathrm{1}},{\mathit{c}}_{\mathrm{2}},\cdots ,{\mathit{c}}_k$ are called a basis of $C$ if they are $\mathit{ℛ}$-independent and generate $C$. Let ${\mathit{G}}_C$ be a $k\times n$ matrix where ${\mathit{c}}_{\mathrm{1}},{\mathit{c}}_{\mathrm{2}},\cdots ,{\mathit{c}}_k$ are rows of ${\mathit{G}}_C$. Then ${\mathit{G}}_C$ is a generator matrix of $C$.

Definition 3   A parity-check matrix ${\mathit{H}}_D$ for a linear code $D$ over $\mathit{ℛ}$ is a generator matrix for the dual code $D^{\perp }$.

Let ${\mathit{M}}_{m\times l}\left(\mathit{ℛ}\right)$ be the set of all $m\times l$ matrices over $\mathit{ℛ}$. For $\mathit{A}\in {\mathit{M}}_{m\times l}\left(\mathit{ℛ}\right)$, ${\mathit{A}}^{\mathrm{T}}$ denotes the transpose of the matrix $\mathit{A}$. We also let $\mathrm{0}$ denote the zero matrix, where the size will either be obvious from the context or specified whenever necessary. Similarly, we denote the $m\times m$ identity matrix by ${\mathit{I}}_m$, or simply $\mathit{I}$ if the size is clear from the context.

Let $\mathit{A}\in {\mathit{M}}_{m\times l}\left(\mathit{ℛ}\right)$, and let ${\mathit{A}}^{\mathrm{r}\mathrm{o}\mathrm{w}}\subset {\mathit{ℛ}}^l$ and ${\mathit{A}}^{\mathrm{c}\mathrm{o}\mathrm{l}}\subset {\mathit{ℛ}}^m$ be the submodules generated by the rows of $\mathit{A}$ and the columns of $\mathit{A}$, respectively. Now, we introduce the definition of the row-rank (or column-rank) of the matrix from Ref. [31].

Definition 4   The parameter $\mathrm{l}\mathrm{o}{\mathrm{g}}_{\left|\mathit{ℛ}\right|}\left|{\mathit{A}}^{\mathrm{r}\mathrm{o}\mathrm{w}}\right|$ is called the row-rank of the matrix $\mathit{A}$ and denoted by $\mathrm{r}{\mathrm{k}}_{\mathrm{r}}(\mathit{A})$, and similarly $\mathrm{l}\mathrm{o}{\mathrm{g}}_{\left|\mathit{ℛ}\right|}\left|{\mathit{A}}^{\mathrm{c}\mathrm{o}\mathrm{l}}\right|$ is called the column-rank of the matrix $\mathit{A}$ and denoted by $\mathrm{r}{\mathrm{k}}_{\mathrm{c}}(\mathit{A})$.

Obviously, when $\mathit{ℛ}$ is a finite field, the above definition coincides with the usual rank of a matrix. We need the following two lemmas which can be found in Ref. [31].

Lemma 5   Let $\mathit{A}\in {\mathit{M}}_{m\times l}\left(\mathit{ℛ}\right)$. Then $\mathrm{r}{\mathrm{k}}_{\mathrm{r}}(\mathit{A})=\mathrm{r}{\mathrm{k}}_{\mathrm{c}}(\mathit{A})$.

In $\mathit{ℛ}$, we define $\mathrm{r}{\mathrm{k}}_{\mathrm{r}}(\mathit{A})$ or $\mathrm{r}{\mathrm{k}}_{\mathrm{c}}(\mathit{A})$ as the rank of the matrix $\mathit{A}$, denoted by $\mathrm{r}\mathrm{k}(\mathit{A})$. The following two concepts and a result about matrices over finite chain rings appear in Ref. [32].

Let $\mathit{A}=\left(a_{ij}\right)\in {\mathit{M}}_{m\times m}\left(\mathit{ℛ}\right)$. If there is an $m\times m$ matrix $\mathit{B}$ over $\mathit{ℛ}$ such that $\mathit{A}\mathit{B}=\mathit{B}\mathit{A}=\mathit{I}$, then $\mathit{A}$ is invertible and $\mathit{B}$ is an inverse of $\mathit{A}$. If the determinant $\mathrm{d}\mathrm{e}\mathrm{t}(\mathit{A})$ is a unit of $\mathit{ℛ}$, then $\mathit{A}$ is non-singular.

Lemma 6   Let $\mathit{A}$ be an $m\times m$ matrix over $\mathit{ℛ}$. The following statements are equivalent: (1) $\mathit{A}$ is invertible; (2) $\mathit{A}$ is non-singular; (3) $\mathrm{r}\mathrm{k}(\mathit{A})=m$.

Lemma 7   Let $\mathit{A}\in {\mathit{M}}_{m\times l}\left(\mathit{ℛ}\right)$ and $\mathit{B}\in {\mathit{M}}_{l\times s}\left(\mathit{ℛ}\right)$. Then $\mathrm{r}\mathrm{k}(\mathit{A}\mathit{B})\le \mathrm{m}\mathrm{i}\mathrm{n}\left\{\mathrm{r}\mathrm{k}(\mathit{A}),\mathrm{r}\mathrm{k}(\mathit{B})\right\}$.

Corollary 1   Let $\mathit{A}\in {\mathit{M}}_{m\times l}\left(\mathit{ℛ}\right)$. If $\mathit{P}\in {\mathit{M}}_{m\times m}$ and $\mathit{Q}\in {\mathit{M}}_{l\times l}$ are non-singular, then $\mathrm{r}\mathrm{k}(\mathit{A})=\mathrm{r}\mathrm{k}(\mathit{P}\mathit{A})=\mathrm{r}\mathrm{k}(\mathit{A}\mathit{Q})=\mathrm{r}\mathrm{k}(\mathit{P}\mathit{A}\mathit{Q})$.

Proof   Let $\mathit{B}=\mathit{P}\mathit{A}$. Then, by Lemma 7, we have $\mathrm{r}\mathrm{k}(\mathit{B})=\mathrm{r}\mathrm{k}(\mathit{P}\mathit{A})\le \mathrm{r}\mathrm{k}(\mathit{A})$. On the other hand, considering the matrix $\mathit{P}$ is non-singular, we obtain $\mathit{A}={\mathit{P}}^{-\mathrm{1}}\mathit{B}$. Again by Lemma 7, we have $\mathrm{r}\mathrm{k}(\mathit{A})=\mathrm{r}\mathrm{k}({\mathit{P}}^{-\mathrm{1}}\mathit{B})\le \mathrm{r}\mathrm{k}(\mathit{B})$. Therefore, $\mathrm{r}\mathrm{k}(\mathit{A})=\mathrm{r}\mathrm{k}(\mathit{P}\mathit{A})$.

Similarly, we can show that $\mathrm{r}\mathrm{k}(\mathit{A})=\mathrm{r}\mathrm{k}(\mathit{A}\mathit{Q})=\mathrm{r}\mathrm{k}(\mathit{P}\mathit{A}\mathit{Q})$.

Definition 5   Let $\mathit{G}$ be a generator matrix of a linear code $C$ over $\mathit{ℛ}$. Then the rank of the code $C$, denoted by $\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C\right)$, is defined as $\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C\right)=\mathrm{r}\mathrm{k}(\mathit{G})$.

Let $C$ be a code of length $n$ over $\mathit{ℛ}$. We define $\overline{C}=\{\overline{\mathit{c}}|\mathit{c}\in C\}$ and $\left(C:r\right)=$$\left\{\mathit{a}\in {\mathit{ℛ}}^n|r\mathit{a}\in C\right\}$, where $r$ is an element of $\mathit{ℛ}$. The following two definitions can be found in Ref. [22].

Definition 6   To any code $C$ over $\mathit{ℛ}$, we associate the tower of codes $C=\left(C:{\gamma }^{\mathrm{0}}\right)\subseteq \left(C:\gamma \right)\subseteq \dots \subseteq \left(C:{\gamma }^{t-\mathrm{1}}\right)$ over $\mathit{ℛ}$ and its projection to ${\mathbb{F}}_q$ ,

$\overline{C}=\overline{(C:{\gamma }^{\mathrm{0}})}\subseteq \overline{(C:\gamma )}\subseteq \dots \subseteq \overline{(C:{\gamma }^{t-\mathrm{1}})}.$

Definition 7   Let $C$ be a linear code over $\mathit{ℛ}$. A generator matrix $\mathit{G}$ for $C$ is said to be in standard form if, after a suitable permutation of the coordinates, $\mathit{G}$ can be written as the following block matrix:

$\mathit{G}=\left(\begin{array}{ccccccc}{\mathit{I}}_{k_{\mathrm{0}}}& {\mathit{A}}_{\mathrm{0,1}}& {\mathit{A}}_{\mathrm{0,2}}& {\mathit{A}}_{\mathrm{0,3}}& \cdots & {\mathit{A}}_{\mathrm{0},t-\mathrm{1}}& {\mathit{A}}_{\mathrm{0},t}\\ \mathrm{0}& \gamma {\mathit{I}}_{k_{\mathrm{1}}}& \gamma {\mathit{A}}_{\mathrm{1,2}}& \gamma {\mathit{A}}_{\mathrm{1,3}}& \cdots & \gamma {\mathit{A}}_{\mathrm{1},t-\mathrm{1}}& \gamma {\mathit{A}}_{\mathrm{1},t}\\ \mathrm{0}& \mathrm{0}& {\gamma }^{\mathrm{2}}{\mathit{I}}_{k_{\mathrm{2}}}& {\gamma }^{\mathrm{2}}{\mathit{A}}_{\mathrm{2,3}}& \cdots & {\gamma }^{\mathrm{2}}{\mathit{A}}_{\mathrm{1},t-\mathrm{1}}& {\gamma }^{\mathrm{2}}{\mathit{A}}_{\mathrm{2},t}\\ ⋮& ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \cdots & {\gamma }^{t-\mathrm{1}}{\mathit{I}}_{k_{t-\mathrm{1}}}& {\gamma }^{t-\mathrm{1}}{\mathit{A}}_{t-\mathrm{1},t}\end{array}\right)=\left(\begin{array}{c}{\mathit{A}}_{\mathrm{0}}\\ \gamma {\mathit{A}}_{\mathrm{1}}\\ ⋮\\ {\gamma }^{t-\mathrm{1}}{\mathit{A}}_{t-\mathrm{1}}\end{array}\right).$(1)

We associate the following matrix with $\mathit{G}$

$\mathit{A}=\left(\begin{array}{c}{\mathit{A}}_{\mathrm{0}}\\ {\mathit{A}}_{\mathrm{1}}\\ ⋮\\ {\mathit{A}}_{t-\mathrm{1}}\end{array}\right).$(2)

For any constant $r\in \mathit{ℛ}$ and any $\mathit{c}\in {\mathit{ℛ}}^n$, we denote by $r\mathit{c}$ the usual multiplication of a vector by a scalar. We say that a vector $\mathit{c}\in {\mathit{ℛ}}^n$ is divisible by a constant $r\in \mathit{ℛ}$, and write as $r\mathit{c}$, if there exists a vector $\mathit{a}\in {\mathit{ℛ}}^n$ such that $\mathit{c}=r\mathit{a}$, i.e., all entries of $\mathit{c}$ are divisible by $r$.

Let $C$ be a linear code over $\mathit{ℛ}$. For $i=\mathrm{1,2},\dots ,t-\mathrm{1},$ we denote by $k_i$ the number of row vectors of $\mathit{G}$ that are divisible by ${\gamma }^i$ but not by ${\gamma }^{i+\mathrm{1}}$. A code with generator matrix of this form is said to have type $\left\{k_{\mathrm{0}},k_{\mathrm{1}},\dots ,k_{t-\mathrm{1}}\right\}$. It is immediate that the number of elements in a code $C$ with this generator matrix is

$\left|C\right|={\left|\mathit{ℛ}/\mathit{m}\right|}^{{\displaystyle \sum _{i=\mathrm{0}}^{t-\mathrm{1}}}\left(t-i\right)k_i}={\left|{\mathbb{F}}_q\right|}^{{\displaystyle \sum _{i=\mathrm{0}}^{t-\mathrm{1}}}\left(t-i\right)k_i}$

Thus, we have $\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C\right)=\frac{\mathrm{1}}{t}{\displaystyle \sum _{i=\mathrm{0}}^{t-\mathrm{1}}}\left(t-i\right)k_i$. This means that the rank of a linear code over $\mathit{ℛ}$ could be a fraction.

2 Constant Rank Codes over Chain Rings

In this section, we first generalize the constant dimension and orbit codes over finite fields to the constant rank and orbit codes over finite chain rings. Then, a method of constructing the constant rank code over $\mathit{ℛ}$ is given.

In order to give the definition of the rank distance of a submodule code over finite chain rings, we need the following lemma.

Lemma 8   Let $C$ and $D$ be two linear codes of length $n$ over $\mathit{ℛ}$. Then we have

$\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C+D\right)=\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C\right)+\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(D\right)-\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C\bigcap D\right)$

Proof   Let $\mathit{u}\in C+D$. Then there are $\mathit{a}\in C,\mathit{b}\in D$ such that $\mathit{u}=\mathit{a}+\mathit{b}$. Obviously, for any $\mathit{w}\in C\bigcap D$, we have

$\mathit{u}=(\mathit{a}+\mathit{w})+(\mathit{b}-\mathit{w}).$

Thus, there are $\left|C\bigcap D\right|$ such expressions for $\mathit{u}$. This means that $\left|C+D\right|=\frac{\left|C\right|\cdot \left|D\right|}{\left|C\bigcap D\right|}$. So,

$\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C+D\right)=\mathrm{l}\mathrm{o}{\mathrm{g}}_{\left|\mathit{ℛ}\right|}\left|C+D\right|=\mathrm{l}\mathrm{o}{\mathrm{g}}_{\left|\mathit{ℛ}\right|}\left|C\right|+\mathrm{l}\mathrm{o}{\mathrm{g}}_{\left|\mathit{ℛ}\right|}\left|D\right|-\mathrm{l}\mathrm{o}{\mathrm{g}}_{\left|\mathit{ℛ}\right|}\left|C\bigcap D\right|$

i.e., $\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C+D\right)=\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C\right)+\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(D\right)-\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(C\bigcap D\right)$.

Definition 8   Let $\mathcal{C}$ be a set of submodules (or linear codes) of length $n$ over $\mathit{ℛ}$. Then $\mathcal{C}$ is called a submodule code of length $n$ over $\mathit{ℛ}$. The submodule code $\mathcal{C}$ is called a constant rank code if all submodules have the same rank. If every submodule of $\mathcal{C}$ has the same type $\left\{k_{\mathrm{0}},k_{\mathrm{1}},\dots ,k_{t-\mathrm{1}}\right\}$, then the submodule code $\mathcal{C}$ is called a constant rank code of type $\left\{k_{\mathrm{0}},k_{\mathrm{1}},\dots ,k_{t-\mathrm{1}}\right\}$.

Definition 9   Let $\mathcal{u}$ and $\mathcal{v}$ be two submodules over $\mathit{ℛ}$. Then the rank distance is defined as

$d_M\left(\mathcal{u},\mathcal{v}\right):=\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}+\mathcal{v}\right)-\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}\bigcap \mathcal{v}\right).$

The minimum rank distance of a submodule code $\mathcal{C}$ is definite as

$d_M\left(\mathcal{C}\right):=\mathrm{m}\mathrm{i}\mathrm{n}\left\{d_M\left(\mathcal{u},\mathcal{v}\right)|,\mathcal{u}\ne \mathcal{v},\mathcal{u},\mathcal{v}\in \mathcal{C}\right\}.$

Remark 1   By Lemma 8, we have

$d_M\left(\mathcal{u},\mathcal{v}\right)=\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}\right)+\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{v}\right)-\mathrm{2}\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}\bigcap \mathcal{v}\right)=\mathrm{2}\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}+\mathcal{v}\right)-\left(\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}\right)+\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{v}\right)\right).$

As a consequence, suppose that $\mathcal{C}$ is a constant rank code of rank $k$ and length $n$, then its minimum submodule distance is even integer and it is upper bounded by

$d_M\left(\mathcal{C}\right)\le \{\begin{array}{l}\mathrm{2}k\mathrm{i}\mathrm{f}\mathrm{2}k\le n,\\ \mathrm{2}\left(n-k\right)\mathrm{i}\mathrm{f}\mathrm{2}k\ge n.\end{array}$

This bound is attained by submodule code in which the intersection of every two different submodules has rank of max{0, 2k$-$n} .

Remark 2   When $\mathit{ℛ}={\mathbb{F}}_q$, the submodule code $\mathcal{C}$ is a subspace code. In this case, the distance between two subspace is $d_M\left(\mathcal{u},\mathcal{v}\right)=d_S\left(\mathcal{u},\mathcal{v}\right)=\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\left(\mathcal{u}\right)+\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\left(\mathcal{v}\right)-\mathrm{2}\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\left(\mathcal{u}\bigcap \mathcal{v}\right)$. This means that the distance of subspace code over ${\mathbb{F}}_q$ is a special case of the rank distance of submodule code over $\mathit{ℛ}$.

Let $\mathcal{C}$ be a constant rank code of rank $k$ and length $n$ over $\mathit{ℛ}$. If it attains bound $d_M\left(\mathcal{C}\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\mathrm{2}k,\mathrm{2}\left(n-k\right)\right\}$, then $\mathcal{C}$ is called an optimum distance constant rank code.

A code $\mathcal{C}$ is called an ${\left(n,\left|\mathcal{C}\right|,d;K\right)}_q$ submodule code over $\mathit{ℛ}$ if the ranks of the codewords of $\mathcal{C}$ are contained in a set $K\subset \{\mathrm{0},\mathrm{1},\mathrm{2},\cdots ,n\}$. In the case $K=\left\{k\right\}$, i.e., $\mathcal{C}$ is a constant rank code, we denote its parameters by ${(n,\left|\mathcal{C}\right|,d;k)}_q$, where $q$ is the number of elements of the residue field ${\mathbb{F}}_q$. If all codewords do not have the same rank, then $\mathcal{C}$ is called a mixed rank code. Such submodule code is denoted by ${\left(n,\left|\mathcal{C}\right|,d\right)}_q$. Constant rank codes are the most well-studied submodule codes, being the analogues of classical codes over finite rings.

Remark 3   When $\mathit{ℛ}={\mathbb{F}}_q$, the submodule code $\mathcal{C}$ is a subspace code. In this case, the distance between two subspace is $d_M(\mathcal{u},\mathcal{v})=d_S(\mathcal{u},\mathcal{v})$. This means that the parameters of submodule codes over $\mathit{ℛ}$ are generalizations of the parameters of subspace codes over ${\mathbb{F}}_q$.

In order to connect a constant rank code of rank $k$ and length $n$ over $\mathit{ℛ}$ with a constant dimension code of dimension $k$ and length $n$ over ${\mathbb{F}}_q$, we need the following lemma, which can be found in Ref. [19].

Lemma 9   (Lemma 9 in Ref. [19]) Suppose that C is a linear code of length $n$ over $\mathit{ℛ}$ with a generator matrix $\mathit{G}$ in standard form (1) and let $A$ be as in $(\mathrm{2})$. Then, for $\mathrm{0}\le i\le t-\mathrm{1}$, $\overline{\left(C:{\gamma }^i\right)}$ has a generator matrix

$\overline{{\mathit{G}}_i}=\left(\overline{\begin{array}{c}{\mathit{A}}_{\mathrm{0}}\\ ⋮\\ \overline{{\mathit{A}}_i}\end{array}}\right)$

In addition, $\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathbb{F}}\overline{(C:{\gamma }^i)}=k_{\mathrm{0}}+k_{\mathrm{1}}+\cdots +k_i$.

Lemma 10   Let $C$ and $D$ be two linear codes over $\mathit{ℛ}$. Then, for $i=\mathrm{0,1},\dots ,t-\mathrm{1}$, we have

$\overline{\left(C\bigcap D:{\gamma }^i\right)}\subset \overline{\left(C:{\gamma }^i\right)}\bigcap \overline{\left(D:{\gamma }^i\right)}.$(3)

$\overline{\left(C:{\gamma }^i\right)}+\overline{\left(D:{\gamma }^i\right)}\subset \overline{\left(C+D:{\gamma }^i\right)}.$(4)

Proof   1) We first prove that

$\overline{\left(C\bigcap D:{\gamma }^i\right)}=\overline{\left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right)}.$(5)

In fact, for any $\mathit{a}\in \left(C\bigcap D:{\gamma }^i\right)$, we have ${\gamma }^i\mathit{a}\in C\bigcap D$. So, ${\gamma }^i\mathit{a}\in C$ and ${\gamma }^i\mathit{a}\in D$, which implies that $\mathit{a}\in \left(C:{\gamma }^i\right)$ and $\mathit{a}\in \left(D:{\gamma }^i\right)$. Thus, $\mathit{a}\in \left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right)$, i. e.,

$\left(C\bigcap D:{\gamma }^i\right)\subset \left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right).$(6)

On the other hand, let $\mathit{b}\in \left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right)$. Then ${\gamma }^i\mathit{b}\in C$ and ${\gamma }^i\mathit{b}\in D$. This means that ${\gamma }^i\mathit{b}\in C\bigcap D$,i. e., $\mathit{b}\in \left(C\bigcap D:{\gamma }^i\right)$. Thus,

$\left(C\bigcap D:{\gamma }^i\right)\supset \left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right).$(7)

Combining Eqs. (6) and (7), we have $\left(C\bigcap D:{\gamma }^i\right)=\left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right)$. So, the Eq. (5) holds.

Next, let $\overline{\mathit{u}}\in \overline{\left(C\bigcap D:{\gamma }^i\right)}=\overline{\left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right)}$, where $\mathit{u}\in \left(C:{\gamma }^i\right)\bigcap \left(D:{\gamma }^i\right)$. Then $\mathit{u}\in \left(C:{\gamma }^i\right)$ and $\mathit{u}\in \left(D:{\gamma }^i\right)$. Therefore, $\overline{\mathit{u}}\in \overline{\left(C:{\gamma }^i\right)}$ and $\overline{\mathit{u}}\in \overline{\left(D:{\gamma }^i\right)}$, which implies that $\overline{\mathit{u}}\in \overline{\left(C:{\gamma }^i\right)}\bigcap \overline{\left(D:{\gamma }^i\right)}$. Thus, we complete the proof of Eq. (3).

2) For any $\mathit{w}\in \overline{\left(C:{\gamma }^i\right)}+\overline{\left(D:{\gamma }^i\right)}$, there are $\mathit{u}\in \overline{\left(C:{\gamma }^i\right)}$ and $\mathit{v}\in \overline{\left(D:{\gamma }^i\right)}$ such that $\mathit{w}=\mathit{u}+\mathit{v}$. Thus, we can find that $\mathit{a}\in \left(C:{\gamma }^i\right)$ and $\mathit{b}\in \left(D:{\gamma }^i\right)$ with $\mathit{u}=\overline{\mathit{a}}$ and $\mathit{v}=\overline{\mathit{b}}$. This means that ${\gamma }^i\mathit{a}\in C$ and ${\gamma }^i\mathit{b}\in D$. Therefore, ${\gamma }^i(\mathit{a}+\mathit{b})\in C+D$, i.e., $\mathit{a}+\mathit{b}\in (C+D:{\gamma }^i)$. Hence, $\overline{\mathit{a}+\mathit{b}}\in \overline{(C+D:{\gamma }^i)}$. Since $\stackrel{}{¯}$ is a homomorphism from ${\mathit{ℛ}}^n$ onto ${\mathbb{F}}_q^n$, we have $\mathit{w}=\mathit{u}+\mathit{v}=\overline{\mathit{a}}+\overline{\mathit{b}}=\overline{\mathit{a}+\mathit{b}}\in \overline{(C+D:{\gamma }^i)}$. This prove that $\overline{\left(C:{\gamma }^i\right)}+\overline{\left(D:{\gamma }^i\right)}\subset \overline{(C+D:{\gamma }^i)}$, i.e., we complete the proof of Eq. (4).

Let $\mathcal{C}$ be submodule code of length $n$ over $\mathit{ℛ}$. In the following, we denote by $\overline{\left(\mathcal{C}:{\gamma }^{t-\mathrm{1}}\right)}$ the set $\left\{\overline{(\mathcal{u}:{\gamma }^{t-\mathrm{1}})}|\mathcal{u}\in \mathcal{C}\right\}$.

Combining Lemmas 9 and 10, we give the following theorem.

Theorem 1   Let $\mathcal{C}$ be a submodule code of length $n$ over $\mathit{ℛ}$ with type $\left\{k_{\mathrm{0}},k_{\mathrm{1}},\dots ,k_{t-\mathrm{1}}\right\}$ and the minimum rank distance $d_M\left(\mathcal{C}\right)$. Then

$\mathrm{1})$ $\overline{\left(\mathcal{C}:{\gamma }^{t-\mathrm{1}}\right)}$ is a constant dimension code over ${\mathbb{F}}_q$ of dimension $l=k_{\mathrm{0}}+k_{\mathrm{1}}+\cdots +k_{t-\mathrm{1}}$ and length $n$. In addition,$d_S\left(\overline{\left(\mathcal{C}:{\gamma }^{t-\mathrm{1}}\right)}\right)\le d_M\left(\mathcal{C}\right)$.

$\mathrm{2})$ In particular, write $k=\frac{\mathrm{1}}{t}{\displaystyle \sum _{i=\mathrm{0}}^{t-\mathrm{1}}}(t-i)k_i$. If $\mathrm{2}k\le n$ and $\mathcal{C}$ is an optimum distance constant rank code, then $\overline{\left(\mathcal{C}:{\gamma }^{t-\mathrm{1}}\right)}$ is also an optimum distance constant dimension code with the minimum distance $\mathrm{2}l$.

Proof   1) By assumptions, for any $\mathcal{u},\mathcal{v}\in \mathcal{C}$, we have $\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}\right)=\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{v}\right)=\frac{\mathrm{1}}{t}{\displaystyle \sum _{i=\mathrm{0}}^{t-\mathrm{1}}}(t-i)k_i.$

According to Lemma 9, for any $\overline{(\mathcal{u}:{\gamma }^{t-\mathrm{1}})},\overline{(\mathcal{v}:{\gamma }^{t-\mathrm{1}})}\in \overline{\left(\mathcal{C}:{\gamma }^{t-\mathrm{1}}\right)}$, we obtain $\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\overline{(\mathcal{u}:{\gamma }^{t-\mathrm{1}}})=\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\overline{(\mathcal{v}:{\gamma }^{t-\mathrm{1}}})={\displaystyle \sum _{i=\mathrm{0}}^{t-\mathrm{1}}}{k}_i.$

Thus, $\overline{\left(\mathcal{C}:{\gamma }^{t-\mathrm{1}}\right)}$ is a constant dimension code over ${\mathbb{F}}_q$ of dimension $k_{\mathrm{0}}+k_{\mathrm{1}}+\cdots +k_{t-\mathrm{1}}$ and length $n$.

On the other hand, by Lemmas 9 and 10, for any $\mathcal{u},\mathcal{v}\in \mathcal{C}$, we obtain

$d_M(\mathcal{u},\mathcal{v})=\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}\right)+\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{v}\right)-\mathrm{2}\mathrm{r}\mathrm{a}\mathrm{n}{\mathrm{k}}_{\mathit{ℛ}}\left(\mathcal{u}\bigcap \mathcal{v}\right)=\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\overline{(\mathcal{u}:{\gamma }^{t-\mathrm{1}})}+\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\overline{(\mathcal{v}:{\gamma }^{t-\mathrm{1}})}-\mathrm{2}\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\overline{(\mathcal{u}\bigcap \mathcal{v}:{\gamma }^{t-\mathrm{1}})}\ge \mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\overline{(\mathcal{u}:{\gamma }^{t-\mathrm{1}})}+\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\overline{(\mathcal{v}:{\gamma }^{t-\mathrm{1}})}-\mathrm{2}\mathrm{d}\mathrm{i}{\mathrm{m}}_{{\mathbb{F}}_q}\left(\overline{(\mathcal{u}:{\gamma }^{t-\mathrm{1}})}\bigcap \overline{(\mathcal{v}:{\gamma }^{t-\mathrm{1}})}\right)=d_S\left(\overline{(\mathcal{u}:{\gamma }^{t-\mathrm{1}})},\overline{(\mathcal{v}:{\gamma }^{t-\mathrm{1}})}\right).$