Matroids and Codes with the Rank Metric

TL;DR

This paper explores the connection between q-analogue matroids and rank metric codes, establishing identities that link their combinatorial and algebraic properties, with implications for coding theory.

Contribution

It introduces a Greene type identity for rank generating functions and provides a combinatorial proof of a MacWilliams identity for Delsarte rank-metric codes.

Findings

Proved a Greene type identity for rank generating functions.

Established a MacWilliams type identity for Delsarte rank-metric codes.

Linked matroid theory with rank metric code properties.

Abstract

We study the relationship between a q-analogue of matroids and linear codes with the rank metric in the vector space of matrices with entries in a finite field. We prove a Greene type identity for the rank generating function of these matroidal structures and the rank weight enumerator of these linear codes. As an application, we give a combinatorial proof of a MacWilliams type identity for Delsarte rank-metric codes.

Equations

$$\rho (X):=\dim C\setminus (S-X)(=\dim C-\dim C(S-X)),$$

$$C(Y):=\{\mathbf{x}\in C:\text{mbox}\mathrm{supp}(\mathbf{x})\subseteq Y\}.$$

$$R(M;x,y):=\sum _{X\subseteq S}{x}^{\rho (S)-\rho (X)}{y}^{|X|-\rho (X)}.$$

$$W_C(x,y)=y^{n-\dim C}(x-y{)}^{\dim C}R(M_C;\frac{qy}{x-y},\frac{x-y}{y}).$$

$$\mathcal{C}(J)$$

$$J^{\perp }$$

$$\mathbf{x}$$

$$\rho (J):={\dim }_{{\mathbb{F}}_q}\mathcal{C}-{\dim }_{{\mathbb{F}}_q}\mathcal{C}(J^{\perp }),$$

$$\dim \mathcal{C}(J_1^{\perp })+\dim \mathcal{C}(J_2^{\perp })=\dim \mathcal{C}(J_1^{\perp })\cap \mathcal{C}(J_2^{\perp })+\dim (\mathcal{C}(J_1^{\perp })+\mathcal{C}(J_2^{\perp })).$$

$$\mathcal{C}(J_1^{\perp })\cap \mathcal{C}(J_2^{\perp })=\mathcal{C}(J_1^{\perp }\cap J_2^{\perp })=\mathcal{C}((J_1+J_2{)}^{\perp }),$$

$$\mathcal{C}(J_1^{\perp })+\mathcal{C}(J_2^{\perp })\subseteq \mathcal{C}(J_1^{\perp }+J_2^{\perp })=\mathcal{C}((J_1\cap J_2{)}^{\perp }),$$

$$\rho (J_1)+\rho (J_2)$$

$$R_P(X_1,X_2,X_3,X_4):=\sum _{D\in \Sigma (E)}{f}_P^D(X_1,X_2)g^{\dim D}(X_3,X_4),$$

$$f_P^J(X,Y):=X^{\rho (E)-\rho (J)}{Y}^{r\dim J-\rho (J)},$$

$$g^l(X,Y):=\prod _{i=0}^{l-1}(X-q^{i}Y),$$

$$0\le \rho (A)-\rho (B)-r(\dim A-\dim B).$$

$$\rho (D)\le r\dim D=r(\dim B-\dim A).$$

$$\rho (A)+\rho (D)\ge \rho (A+D)+\rho (A\cap D)\ge \rho (B).$$

$${\rho }^{\ast }(J):=\rho (J^{\perp })+r\dim J-\rho (E).$$

$${\rho }^{\ast }(J_2)-{\rho }^{\ast }(J_1)$$

$${\rho }^{\ast }(J_1\cap J_2)$$

$${\rho }^{\ast }(J_1+J_2)$$

$${\rho }^{\ast }(J_1)$$

$${\rho }^{\ast }(J_2)$$

$${\rho }^{\ast }(J_1\cap J_2)+{\rho }^{\ast }(J_1+J_2)-{\rho }^{\ast }(J_1)-{\rho }^{\ast }(J_2)$$

$${\hat{R}}_P(X_1,X_2,X_3,X_4):=\sum _{D\in \Sigma (E)}{f}_P^D(X_1,X_2)g^{\dim D^{\perp }}(X_3,X_4).$$

$$R_{P^{\ast }}(X_1,X_2,X_3,X_4)={\hat{R}}_P(X_2,X_1,X_3,X_4).$$

$$R_{P^{\ast }}(X_1,X_2,X_3,X_4)$$

$$⟨M,N⟩:=\text{mbox}\mathrm{Tr}(M{N}^{\text{mbox}\mathrm{T}}),$$

$$⟨M,N⟩=\sum _{i=1}^m{M}_i\cdot N_i,$$

$${\cal C}^{\perp}:=\{N\in\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})\>:\>\mbox{\rm$\langle M,N\rangle=0$ for all $M\in{\cal C}$}\}.$$

$$(\mathcal{V}(R){)}^{\perp }=\mathcal{V}(R^{\perp }).$$

$$⟨M,N⟩$$

Full text

Keywords: rank-metric code; matroid; polymatroid; Greene’s identity; MacWilliams identity

Mathematics Subject Classification: 05B35, 94B60

1 Introduction

Let $S$ be a finite set and let $\rho\>:\>2^{S}\rightarrow{\mathbb{Z}}$ be a function. An ordered pair $M=(S,\rho)$ is called a matroid if

1. (1)

If $X\subseteq S$, then $0\leq\rho(X)\leq|X|$. 2. (2)

If $X,Y\subseteq S$ and $X\subseteq Y$, then $\rho(X)\leq\rho(Y)$. 3. (3)

If $X,Y\subseteq S$, then $\rho(X\cup Y)+\rho(X\cap Y)\leq\rho(X)+\rho(Y).$

Matroids are constructed from a lot of combinatorial or algebraic structures including graphs and matrices (cf. [29, 20]). Let ${\mathbb{F}}_{q}$ be a finite filed of $q$ elements. For an $[n,k]$ code $C$ over ${\mathbb{F}}_{q}$ and any subset $X\subseteq S:=\{1,2,\ldots,n\}$, define

[TABLE]

where, for any $Y\subseteq S$, $C\setminus Y$ denotes the punctured code by $Y$ and

[TABLE]

Then we find that $M_{C}:=(S,\rho)$ satisfies the above conditions and so $M_{C}$ is a matroid.

The (Whitney) rank generating function of a matroid $M=(S,\rho)$ is defined by

[TABLE]

In 1976, Greene ([14]) proved one celebrated result, known as Greene’s identity, on the relationship between the Hamming weight enumerator $W_{C}(x,y)$ of an $[n,k]$ code $C$ over ${\mathbb{F}}_{q}$ and the rank generating function of the corresponding matroid $M_{C}$ as follows:

[TABLE]

As an application of this result, he gave a elegant combinatorial proof of the MacWilliams identity for the Hamming weight enumerator (cf. [29]). After this paper, a number of paper in the link on matroids and codes have been published, for instance, generalizations of the identity to higher weight enumerators ([2, 3, 25]), and to some enumerators for other classes of codes ([5, 6, 26, 28]). In particular, Britz ([3]) proved the equivalence between the set of higher weight enumerators of a linear code over a finite filed and the rank generating function associated to the code. In [4] and [16], the authors studied the critical problem, known as a classical problem in matroid theory, from coding theoretical approach.

Delsarte introduced a rank-metric code as a set of matrices of given size over a finite field in [8]. He mainly studied the rank-metric code based on his research on association schemes. In [9, 10], Gabidulin gave a different definition of rank-metric codes such as a set of vectors in a vector space over an extension field. The rank-weight of a codeword is defined by the rank of the associated matrix. Ravagnani proved that Gabidulin codes can be regarded as a special case of Delsarte codes and he compared the duality theories of these codes in [22]. Some useful applications of rank-metric codes are widely known, for instance, to space-time codes ([27]), to network coding ([24]), and to cryptography ([11]).

In addition, some of the results in classical coding theory are generalized to rank-metric codes. For instance, a Singleton bound on the minimum rank distance of the codes are proven in [8]. Some constructions of the codes which attain the bound, called maximum rank distance (MRD) codes, are known ([7, 8, 9, 10, 17]). The MacWilliams type identitis for rank weight enumerators of rank-metric codes are proposed as several expressions: an algebraic expression ([8]), identitis for rank weight enumerators of Gabidulin rank-metric codes closed to the original form ([12, 13]), and moments of rank distributions of Delsarte rank-metric codes ([22]).

The layout of this paper is as follows. In Section 2, we firstly introduce some basic notion on Delsarte rank-metric codes and new matroidal structures related rank-metric codes. And we give a Greene type identity for rank-metric codes. Section 3 contains an application of the Greene type identity to the MacWilliams identity for Delsarte rank-metric codes.

Most of the basic terminology in coding theory and matroid theory used in this paper will be found in standard texts (cf. [15, 19, 20, 29]).

2 $(q,r)$-Polymatroids and Rank Generating Functions

We denote by $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$ the ${\mathbb{F}}_{q}$-vector space of $n\times m$ matrices with entries in ${\mathbb{F}}_{q}$. A Delsarte rank-metric code ${\cal C}$ of size $n\times m$ over ${\mathbb{F}}_{q}$ is an ${\mathbb{F}}_{q}$-linear subspace of $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. Throughout this paper, set $E:={\mathbb{F}}_{q}^{n}$. For convenience, we shall write for $D\leq E$ if $D$ is a subspace of $E$. For any subspace $J\leq E$, define

[TABLE]

where $\mbox{\rm col}(M)$ denotes the column space of $M$ over ${\mathbb{F}}_{q}$.

Lemma 1

${\cal C}(J)$* is an ${\mathbb{F}}_{q}$-linear subspace of $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$.*

**Proof.  **For any $M,N\in{\cal C}(J)$, let ${\bm{u}}_{1},\ldots,{\bm{u}}_{m}$ and ${\bm{v}}_{1},\ldots,{\bm{v}}_{m}$ be the column vectors of $M$ and $N$, respectively. For any $\alpha,\beta\in{\mathbb{F}}_{q}$, take a vector ${\bm{x}}\in\mbox{\rm col}(\alpha M+\beta N)$. Then ${\bm{x}}$ can be written as

[TABLE]

for some $a_{1},\ldots,a_{m}\in{\mathbb{F}}_{q}$. Since $\alpha(a_{1}{\bm{u}}_{1}+\cdots+a_{m}{\bm{u}}_{m})\in J$ and $\beta(a_{1}{\bm{v}}_{1}+\cdots+a_{m}{\bm{v}}_{m})\in J$, we have that ${\bm{x}}\in J$. $\Box$

We denote by $\Sigma(E)$ the set of all subspaces of $E$ and denote by $\Sigma(J)$ the set of all subspaces of any subspace $J\in\Sigma(E)$.

Now we shall introduce a $(q,r)$-polymatroid as a $q$-analogue of $k$-polymatroids (cf. [21]).

Definition 2

A $(q,r)$-polymatroid is an ordered pair $P=(E,\rho)$ consisting of a vector space $E:={\mathbb{F}}_{q}^{n}$ and a function $\rho\>:\>\Sigma(E)\to{\mathbb{Z}}^{+}\cup\{0\}$ having the following properties:

1. (R1)

If $A\leq E$, then $0\leq\rho(A)\leq r\dim A$. 2. (R2)

If $A,B\leq E$ and $A\subseteq B$, then $\rho(A)\leq\rho(B)$. 3. (R3)

If $A,B\leq E$, then $\rho(A+B)+\rho(A\cap B)\leq\rho(A)+\rho(B).$

We find easily that $\rho(\{{\bm{0}}\})=0$ from (R1) and a $(q,1)$-polymatroid is a $q$-analogue of matroids.

Proposition 3

Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$, and let $\rho\>:\>\Sigma(E)\to{\mathbb{Z}}^{+}\cup\{0\}$ be the function defined by

[TABLE]

for any subspace $J$ of $E$. Then $P_{\cal C}:=(E,\rho)$ is a $(q,m)$-polymatroid.

**Proof.  **Clearly the function $\rho$ satisfies (R1) and (R2). Now we prove that (R3) holds. Let $J_{1}$ and $J_{2}$ are subspaces of $E$. From Lemma 1, we have that

[TABLE]

Moreover, it follows that

[TABLE]

because of $\mbox{\rm col}(M_{1}+M_{2})\subseteq\mbox{\rm col}(M_{1})+\mbox{\rm col}(M_{2})$ for any $M_{1},M_{2}\in\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. Therefore we have that

[TABLE]

Thus $P_{{\cal C}}=(E,\rho)$ is a $(q,m)$-polymatroid. $\Box$

The rank generating function of a $(q,r)$-polymatroid $P=(E,\rho)$ is defined by

[TABLE]

where

[TABLE]

for any subspace $J\in\Sigma(E)$, and

[TABLE]

for any nonnegative integer $l\in{\mathbb{Z}}^{+}\cup\{0\}$.

The following lemma is essential.

Lemma 4

Let $P=(E,\rho)$ be a $(q,r)$-polymatroid. If $A$ and $B$ are subspaces of $E$ such as $A\subseteq B$, then

[TABLE]

**Proof.  **Let $D$ be a subspace of $E$ generated by $B-A$. From (R1), we have that

[TABLE]

From (R3), we also have that

[TABLE]

By combining the above two inequalities, the lemma follows. $\Box$

Proposition 5

For any $(q,r)$-polymatroid $P=(E,\rho)$ and any subspace $J$ of $E$, define

[TABLE]

Then $P^{*}=(E,\rho^{*})$ is a $(q,r)$-polymatroid.

Proof.  $J^{\perp}\subseteq E$ implies that $\rho(J^{\perp})\leq\rho(E)$. So we have that $(0\leq)\rho^{*}(J)\leq r\dim J$. Thus (R1) holds.

Let $J_{1}$ and $J_{2}$ are subspaces of $E$. Assume that $J_{1}\subseteq J_{2}$. Then it follows from Lemma 4 that

[TABLE]

Hence (R2) holds.

From the definition of $\rho^{*}$, it follows that

[TABLE]

From the inequality in (R3) for $\rho$, we have that

[TABLE]

Then (R3) holds. $\Box$

For a rank generating function $R_{P}(X_{1},X_{2},X_{3},X_{4})$ of a $(q,r)$-polymatroid $P=(E,\rho)$, define

[TABLE]

The following result is a duality on rank generating functions of $(q,r)$-polymatroids.

Theorem 6

Let $P=(E,\rho)$ be a $(q,r)$-polymatroid. Then

[TABLE]

**Proof.  **It follows from the definition of rank generating functions that

[TABLE]

$\Box$

The trace product of matrices $M,N\in\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$ is defined by

[TABLE]

where Tr denotes the trace of a matrix. Then it is easy to find that the trace product can be calculate from the standard inner products of each row vectors as follows:

Lemma 7

Let $M_{i}$ and $N_{i}$ be the $i$-th column vectors of $M$ and $N$, respectively. Then we have that

[TABLE]

where $M_{i}\cdot N_{i}$ denotes the standard inner product of vectors $M_{i}$ and $N_{i}$.

Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. The dual code of ${\cal C}$ is defined by

[TABLE]

Then the following result is well-known (see, Lemma 5 in [22]).

Lemma 8

([22])* For any Delsarte rank-metric codes ${\cal C},{\cal D}\in\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$, it follows that*

1. (1)

$({\cal C}^{\perp})^{\perp}={\cal C}$; 2. (2)

$\dim_{{\mathbb{F}}_{q}}({\cal C}^{\perp})=nm-\dim_{{\mathbb{F}}_{q}}({\cal C})$; 3. (3)

$({\cal C}\cap{\cal D})^{\perp}={\cal C}^{\perp}+{\cal D}^{\perp}$, and $({\cal C}+{\cal D})^{\perp}={\cal C}^{\perp}\cap{\cal D}^{\perp}$.

Set ${\cal V}:=\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. Then it turns out easily that, for any subspace $R$ of ${\mathbb{F}}_{q}^{n}$,

1. (1)

${\cal C}(R)={\cal V}(R)\cap{\cal C}$, for any Delsarte rank-metric code ${\cal C}\subseteq{\cal V}$; 2. (2)

$|{\cal V}(R)|=|R|^{m}=q^{m\dim R}$.

Lemma 9

For any subspace $R$ of ${\mathbb{F}}_{q}^{n}$,

[TABLE]

**Proof.  **For any matrix $M\in{\cal V}(R^{\perp})$, Lemma 7 implies that

[TABLE]

for all $N\in{\cal V}(R)$. Thus we have that $M\in({\cal V}(R))^{\perp}$ and so ${\cal V}(R^{\perp})\subseteq({\cal V}(R))^{\perp}$.

Conversely, take a matrix $M\in({\cal V}(R))^{\perp}$. Set

[TABLE]

Let $M({\bm{x}}_{i},j)$ be the $n\times m$ matrix over ${\mathbb{F}}_{q}$ with ${\bm{x}}_{i}$ as the $j$-th column vector and ${\bm{0}}$ elsewhere. Then it turns out that $M({\bm{x}}_{i},j)\in{\cal V}(R)$. Thus we have that

[TABLE]

This implies that $M_{j}\in R^{\perp}$, for all $j=1,2,\ldots,m$. Therefore it follows that $M\in{\cal V}(R^{\perp})$ and so ${\cal V}(R^{\perp})\supseteq({\cal V}(R))^{\perp}$. $\Box$

Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. The dual space ${\cal C}^{\star}$ of ${\cal C}$ is defined by

[TABLE]

Since ${\cal C}$ is a finite free ${\mathbb{F}}_{q}$-module, there is a (non-natural) isomorphism ${\cal C}\cong{\cal C}^{\star}$ (see, for instance, [18]).

The following exact sequence is an analogue of the basic exact sequence on Proposition 2 in [23].

Proposition 10

Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. For any subspace $R$ of ${\mathbb{F}}_{q}^{n}$, there is an exact sequence as ${\mathbb{F}}_{q}$-modules,

[TABLE]

where $f$ is an ${\mathbb{F}}_{q}$-homomorphism defined by

[TABLE]

and the maps inc and res denote the inclusion map and the restriction map, respectively.

**Proof.  **The inclusion map inc is a natural injection, and the restriction map res is surjective, since ${\mathbb{F}}_{q}$ is an injective module over itself.

Take $M\in{\cal C}^{\perp}(R)$. Then it follows that

[TABLE]

for all $N\in{\cal C}$. This implies that $M\in\ker f$ and so $\mbox{\rm Im}(\mbox{\rm inc})\subseteq\ker f$. Conversely, take $M\in\ker f$. Then we have that $\hat{M}(N)=\langle M,N\rangle=0$ for all $N\in{\cal C}$ and so $M\in{\cal C}^{\perp}\cap{\cal V}(R)={\cal C}^{\perp}(R)$. Therefore we have that $\mbox{\rm Im}(\mbox{\rm inc})\supseteq\ker f$. Thus it follows that $\mbox{\rm Im}(\mbox{\rm inc})=\ker f$.

Next we prove that $\ker(\mbox{\rm res})=\mbox{\rm Im}f$. Take $M\in{\cal V}(R)$. Then it also follows from Lemma 7 that $\hat{M}(N)=\langle M,N\rangle=0$ for all $N\in{\cal C}(R^{\perp})$. This implies that $\hat{M}\in\ker(\mbox{\rm res})$ and so $\mbox{\rm Im}f\subseteq\ker(\mbox{\rm res})$. Conversely, take $\lambda\in\ker(\mbox{\rm res})$. The map $f$ is surjective. Thus there exists $M\in{\cal V}$ such that $\lambda=\hat{M}$. Therefore we have that $\hat{M}(N)=\langle M,N\rangle=0$ for all $N\in{\cal C}(R^{\perp})$, and so $M\in({\cal C}(R^{\perp}))^{\perp}$. From Lemma 9, we have that

[TABLE]

and so $M\in{\cal C}^{\perp}+{\cal V}(R)$. Then $M=M_{1}+M_{2}$ for some $M_{1}\in{\cal C}^{\perp}$ and $M_{2}\in{\cal V}(R)$. Since $\hat{M_{1}}(N)=0$ for all $N\in{\cal C}$, it turns out that $\lambda=\hat{M}=\hat{M_{2}}\in\mbox{\rm Im}f$. Thus it follows that $\mbox{\rm Im}f=\ker(\mbox{\rm res})$.

Therefore the proposition follows. $\Box$

Proposition 11

For any Delsarte rank-metric code ${\cal C}$ in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$, let $P_{\cal C}=(E,\rho)$ be the $(q,m)$-polymatroid obtained from ${\cal C}$ such as discussed in Proposition 3. Then $P_{\cal C}^{*}=P_{{\cal C}^{\perp}}$.

**Proof.  **Take any subspace $J\subseteq E$. Set

[TABLE]

From the definition of dual $(q,m)$-polymatroids, we have that

[TABLE]

By applying $J^{\perp}$ for $R$ in Proposition 10, we have that

[TABLE]

From the above equation and Lemma 8(2), it follows that

[TABLE]

$\Box$

Definition 12

(cf. [13, 22]) Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. The rank distribution of ${\cal C}$ is the collection $\{A_{i}({\cal C})\}_{i\in{\mathbb{Z}}_{\geq 0}}$, where

[TABLE]

The rank weight enumerator of ${\cal C}$ is defined by

[TABLE]

For any Delsarte rank-metric code ${\cal C}$ in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$ and any subspace $R\subseteq{\mathbb{F}}_{q}^{n}$, set

[TABLE]

Then it turns out easily that $B_{{\cal C}}(R)=\sum_{T\in\Sigma(R)}A_{{\cal C}}(T)$ and $B_{{\cal C}}(R)=|{\cal C}(R)|$. From the Möbius inversion formula (see, for instance, [1]), we have the following equation.

Lemma 13

[TABLE]

where ${a\choose b}$ denotes the binomial coefficient of integers $a\geq b\geq 0$.

Now we prove our main result in this paper. The following equation is a kind of Greene type identities for $(q,r)$-polymatroids and Delsarte rank-metric codes.

Theorem 14

Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$, and let $P_{\cal C}=(E,\rho)$ be the $(q,m)$-polymatroid obtained from ${\cal C}$ such as discussed in Proposition 3. Then

[TABLE]

**Proof.  **By Lemma 13 and the $q$-binomial Theorem, we obtain that

[TABLE]

$\Box$

3 Application to the MacWilliams Identity

In this section, we shall present a MacWilliams type identity for Delsarte rank-metric codes as an application of Theorem 14.

Proposition 15

Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. Then

[TABLE]

where ${a\brack b}_{q}$ denote the Gaussian binomial coefficient (or $q$-binomial coefficient) of non-nagative integers $a$ and $b$.

**Proof.  **By combining Theorem 14, Proposition 11 and Theorem 6, we have that

[TABLE]

$\Box$

The equation in Proposition 15 can be viewed as a kind of MacWilliams type identities for Delsarte rank-matric codes. Now we prove the equivalence between the above equation and the MacWilliams type identity for Gabidulin rank-metric codes proposed in [12, 13] by using the concepts of $q$-products and $q$-derivative for homogeneous polynomials.

Definition 16

([12, 13]) Let $a(x,y;m)=\sum_{i=0}^{r}a_{i}(m)x^{r-i}y^{i}$ and $b(x,y;m)=\sum_{j=0}^{s}b_{j}(m)x^{s-j}y^{j}$ be two homogeneous polynomials in $x$ and $y$ of degree $r$ and $s$ respectively with coefficients $a_{i}(m)$ and $b_{j}(m)$ for $i,j\geq 0$ in turn are real functions of $m$, and are assumed to be zero unless otherwise specified. The $q$-product $c(x,y;m)$ of $a(x,y;m)$ and $b(x,y;m)$ is defined to be the homogeneous polynomial of degree $(r+s)$ as follows:

[TABLE]

where

[TABLE]

For $n\geq 0$, the $n$-th $q$-power of $a(x,y;m)$ is defined recursively:

[TABLE]

Lemma 17

For any $n\geq 0$ and $(0\leq)l\leq n$,

1. (1)

$(q^{m}y)^{[n]}=(q^{m}y)^{n}$, 2. (2)

$(x-y)^{[l]}*(q^{m}y)^{[n-l]}=(x-y)^{[l]}\times(q^{m}y)^{n-l}$.

**Proof.  **(1) We shall prove this equation by induction on $n$. When $n=0$, it follows that $(q^{m}y)^{[0]}=1=(q^{m}y)^{0}$. Suppose that the equation is true for $n-1$, that is, $(q^{m}y)^{[n-1]}=(q^{m}y)^{n-1}$. Then, from the definition of $q$-products, we have that

[TABLE]

(2) From Lemma 1 in [12], it follows that

[TABLE]

where

[TABLE]

Then we have that

[TABLE]

where

[TABLE]

Therefore it follows that

[TABLE]

$\Box$

Definition 18

([12, 13]) The $q$-transform of a homogeneous polynomial

$a(x,y;m)=\sum_{i=0}^{r}a_{i}(m)x^{r-i}y^{i}$ is defined by the homogeneous polynomial

[TABLE]

The following lemma is essential:

Lemma 19

1. (1)

$\displaystyle{a\brack b}_{q}={a-1\brack b}_{q}+q^{a-b}{a-1\brack b-1}_{q}=q^{b}{a-1\brack b}_{q}+{a-1\brack b-1}_{q};$** 2. (2)

$\displaystyle{a\brack b}_{q}{b\brack c}_{q}={a\brack b-c}_{q}{a-b+c\brack c}_{q};$** 3. (3)

$\displaystyle{a\brack b}_{q}=\sum_{i=0}^{n}q^{i(a-b-n+i)}{n\brack i}_{q}{a-n\brack b-i}_{q},\>\>\>n=0,1,\ldots,a;$** 4. (4)

$\displaystyle{a+b\choose 2}={a\choose 2}+ab+{b\choose 2}.$**

By combining Proposition 15 and Lemmas 17 and 19, we have the following MacWilliams type identity for a rank weight enumerator of a Delsarte rank-metric code.

Theorem 20

Let ${\cal C}$ be a Delsarte rank-metric code in $\mbox{\rm Mat}(n\times m,{\mathbb{F}}_{q})$. Then

[TABLE]

**Proof.  **By Corollary 1 in [13], we have that

[TABLE]

where

[TABLE]

In addition, from Proposition 15, it is sufficient to prove that

[TABLE]

By using Lemma 19, we have that

[TABLE]

$\Box$

Acknowledgement

The author would like to thank Prof. Frederique Oggier for introducing him to the concept of rank-metric codes.