%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{135}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
%%\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\usepackage{xcolor}
\usepackage{subcaption}
\usepackage{todonotes}
\usepackage{cleveref}
\usepackage{lipsum}


%%\numberwithin{equation}{section}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem*{Theorem}{Theorem}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{remarks}[theorem]{Remarks}
\newtheorem{example}[theorem]{Example}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{question}[theorem]{Question}
\newtheorem{notation}[theorem]{Notation}

\newcommand{\M}{\mathcal{M}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\B}{\mathcal{B}}

\def\Z{{\mathbb{Z}}}
\def\R{{\mathbb{R}}}
\def\N{{\mathbb{N}}}
\def\K{{\mathbb{K}}}
\def\F{{\mathbb{F}}}
\def\I{{\mathbb{I}}}


%% define your title in the usual way
\title{Most \texorpdfstring{$q$}{q}-matroids are not representable}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author

\author{\texorpdfstring{Sebastian Degen\thanks{\href{mailto:sdegen@math.uni-bielefeld.de}{sdegen@math.uni-bielefeld.de}. Sebastian Degen is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- SFB-TRR 358/1 2023 -- 491392403 and SPP 2458 -- 539866293.}\addressmark{1} \and Lukas~K\"uhne\thanks{\href{mailto:lkuehne@math.uni-bielefeld.de}{lkuehne@math.uni-bielefeld.de}. Lukas~K\"uhne is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- SFB-TRR 358/1 2023 -- 491392403 and SPP 2458 -- 539866293.}\addressmark{2}}{Sebastian Degen and Lukas~K\"uhne}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Universit\"at Bielefeld, Fakult\"at f\"ur Mathematik, Bielefeld, Germany\\ \addressmark{2}Universit\"at Bielefeld, Fakult\"at f\"ur Mathematik, Bielefeld, Germany}

%% put the date of submission here
\received{13.11.2024} % Intial FPSAC submission:   

%% leave this blank until submitting a revised version
\revised{\today} % Final FPSAC submission: 29.03.2025

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)

\abstract{A $q$-matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These $q$-matroids are motivated by coding theory as the representable $q$-matroids are the ones that stem from rank-metric codes. In this note, we establish a $q$-analogue of Nelson's theorem in matroid theory by proving that asymptotically almost all $q$-matroids are not representable. This answers a question about representable $q$-matroids by Jurrius and Pellikaan strongly in the negative.}

%% put your French abstract here, or comment this out if you don't have one
%%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{$q$-matroids, representability, rank-metric codes.}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want

\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{q_matroid_references.bib}
\appto\bibfont{\setlength{\emergencystretch}{1em}}

%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\hfill{{\em In memory of Kai-Uwe Schmidt.}}

\section{Introduction}

The concept of $q$-matroids was recently introduced by Jurrius and Pellikaan in \cite{jp2018}. Their approach views $q$-matroids as the $q$-analogues of matroids by replacing finite sets with finite dimensional vector spaces over finite fields.
We start by recalling their key definition of $q$-matroids.

In the following let $\cL(E)$ denote the  lattice of subspaces of $E$.


\begin{definition}
    A \textbf{$q$-matroid} $\M$ is a pair $(E,\rho)$ of a finite dimensional vector space $E$ over a finite field $\F_q$, for some prime power $q$ and a function $\rho:\cL(E)\rightarrow\Z_{\geq 0}$, called the \textbf{$q$-rank function} satisfying the following axioms for all subspaces $X,Y\leq E$:
    \begin{enumerate}
        \item  $0\leq \rho(X)\leq \dim(X)$,
        \item  If $X\leq Y$, then $\rho(X)\leq \rho(Y)$ and
        \item  $\rho(X\cap Y)+\rho(X+Y)\leq \rho(X)+\rho(Y)$.
    \end{enumerate}
    %Here and in the following $\cL(E)$ denotes the  lattice of subspaces of $E$.
\end{definition}
These axioms are a direct translation of the usual axioms of a matroid in terms of its rank function to the setting of a vector space and its subspaces.
It should be noted that the $q$-analogue of a matroid was also independently introduced much earlier in Crapo's Ph.D. thesis~\cite{crapo}.

One of the main motivations to study $q$-matroids stems from coding theory, as the representable $q$-matroids arise from rank-metric codes. Recently, several new phenomena about rank-metric codes were discovered by studying $q$-(poly)matroids and their properties, see for instance \cite{gjlr2019,gj2022,gj2023,ab2023_1}. 
On the other hand, studying $q$-matroids as analogues of matroids is also an active field of research.~Significant effort has been invested to define $q$-analogue versions of matroidal concepts, such as $q$-cryptomorphic axioms systems, see \cite{ab2023_2,cj2023,bcj2022}, or a $q$-analogue of the direct sum of matroids, see \cite{cj2022}.
It is however fair to say that in these cases the $q$-analogue is much more involved than the immediate translation of the matroid rank function above.

Jurrius and Pellikaan asked whether all $q$-matroids are representable~\cite{jp2018}.
This question turned out to be too optimistic, as the first examples of non-representable appeared recently: 
Gluesing-Luerssen and Jany introduced a method of translating well-known non-representable matroid such as the V\'amos matroid to the $q$-analogue setting~\cite{gj2022} whereas Ceria and Jurrius found the smallest non-representable $q$-matroid which is of rank $2$ on $\F_2^4$~\cite{cj2022}.

The main result of this paper answers the above question negatively in a strong sense by proving the following asymptotic result on representable $q$-matroids.

\begin{theorem}\label{q_mat_asymptotics}
    Let $n$ be an integer, $\mathcal{R}_{q}(n)$ be the number of representable $q$-matroids and $\mathcal{N}_{q}(n)$ be the number of all $q$-matroids on $\F_q^n$, respectively. Then the ratio $\frac{\mathcal{R}_{q}(n)}{\mathcal{N}_{q}(n)}$ asymptotically vanishes, i.e.
    \[
    \lim_{n\rightarrow\infty}\frac{\mathcal{R}_{q}(n)}{\mathcal{N}_{q}(n)}=0.
    \]
\end{theorem}

This result implies that the portion of representable $q$-matroids tends to $0$ as $n$ goes to infinity. 
This is a $q$-analogue version of a celebrated theorem of Nelson which says that asymptotically almost all matroids are not representable~\cite{n2018}.
To prove this theorem, we provide a lower bound on the number $\mathcal{N}_{q}(n)$ via coding theoretic estimations on constant dimension codes and an upper bound on the number $\mathcal{R}_{q}(n)$ via an algebraic concept named zero patterns.

Our paper is structured as follows. In \Cref{preliminaries} we briefly recap the basic notions of $q$-matroids and rank metric codes, needed for the discussion in the later sections. In \Cref{lower_bound} we explain the concept of constant dimension codes and describe a lower bound on their maximal cardinality. Afterwards, we establish their connection to $q$-matroids which yields the lower bound on the number $\mathcal{N}_{q}(n)$. In \Cref{upper_bound} we define the notion of zero patterns of a polynomial sequence and explain how these are connected to representable $q$-matroids. Subsequently, we derive an upper bound on the number of representable $q$-matroids of rank $k$ in $\F_q^n$. We conclude this section with a discussion about the asymptotic behavior of representable $q$-matroids and the proof of our main result, \Cref{q_mat_asymptotics}.

The following is an extended abstract of our paper \cite{dl2024}, recently published on the arXiv. 

% \section*{Acknowledgments}

% We thank Gianira Nicoletta Alfarano and Charlene Weiß for fruitful discussions.
% We are grateful to Kai-Uwe Schmidt for introducing us to the topic of $q$-matroids.

% Both authors are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- SFB-TRR 358/1 2023 -- 491392403 and SPP 2458 -- 539866293.

\section{Preliminaries}
\label{preliminaries}


\subsection{Notation} Throughout the paper we always think of $E$ as $\F_q^n$ for some prime power $q$ and some integer $n\geq 1$. Moreover, we denote by $\binom{E}{k}_q$ the set of all $k$-dimensional subspaces of $E$. Finally we abbreviate the row space of a matrix $M\in\F_q^{k\times n}$ as $\text{rowspan}(M)\in\cL(\F_q^n)$.


\subsection{\texorpdfstring{$q$}{q}-Binomial coefficients}

 The $k$-dimensional subspaces in $\F_q^n$ for $0\leq k\leq n$, are counted via the \textbf{$q$-binomial coefficient} $\binom{n}{k}_q$. The following bounds on the $q$-binomial coefficient will be crucial for the estimations later on in this paper, see \cite[Lemma 2.1, Lemma 2.2]{i2015}.
% \[
%     \binom{n}{k}_q:=\frac{(q^n-1)\cdots(q^{n-k+1}-1)}{(q-1)\cdots(q^k-1)}.
% \]

% The following bounds on the $q$-binomial coefficient will be crucial for the estimations later on in this paper, see \cite[Lemma 2.1, Lemma 2.2]{i2015}.

\begin{lemma}\label{Estimations_q_binom}
    For $0\leq k\leq n$ and $q\geq 2$, the following holds
        \[
        q^{(n-k)k}\leq\binom{n}{k}_q\leq \frac{111}{32}q^{(n-k)k}.
        \]
\end{lemma}

\subsection{\texorpdfstring{$q$}{q}-Matroids}
\label{q_mats}

In this subsection, we shortly recall the basic notions about $q$-matroids, which we frequently use throughout the article. For more details see for instance \cite{jp2018,bcj2022}.



In the introduction, we recalled the definition of a $q$-matroid.~Given such a $q$-matroid $\M=(E,\rho)$ we are interested in its associated concepts.
We call $\rho(\M):=\rho(E)$ the \textbf{rank} of $\M$. A subspace $X\in\cL(E)$ is called \textbf{independent} if $\rho(X)=\dim X$, otherwise it is called \textbf{dependent}. If an independent space~$B$ satisfies $\rho(B)=\rho(E)$, it is called a \textbf{basis} of $\M$. Finally, we call a subspace $C$ a \textbf{circuit} if it is dependent and all its proper subspaces are independent.

As with usual matroids, there are cryptomorphic characterizations of $q$-matroids by its collection of independent spaces, dependent spaces, bases and circuits, see \cite{jp2018,bcj2022, cj2023}.
Somewhat surprisingly unless in the case of the characterization by rank function, these alternative descriptions are generally more involved than the direct translations of the usual matroidal axioms.

In \Cref{lower_bound} we need a special construction of so-called \textbf{paving} $q$-matroids, which are $q$-matroids~$\M$ such that every circuit $C$ in $\M$ satisfies $\dim(C)\geq \rho(\M)$.
One way to obtain paving $q$-matroids is via the following construction, described in~\cite{gj2022}.

\begin{proposition}[{\cite[Proposition 4.5]{gj2022}}]
    Let $n$ be an integer and fix $1\leq k\leq n-1$. Further let $\mathcal{S}$ be a collection of $k$-dimensional subspaces of $\F_q^n$ such that for every two distinct $V,W\in\mathcal{S}$, $\dim(V\cap W)\leq k-2$. Define the map
    
    $$\rho:\cL(E)\rightarrow\mathbb{Z}_{\geq 0},\quad V\mapsto
    \begin{cases}
        \;\hfil k-1\;&\emph{if }V\in\mathcal{S},\\
        \;\hfil \min\{\dim V,k\}\;&\emph{otherwise}. 
    \end{cases}$$
    Then $(E,\rho)$ is a paving $q$-matroid of rank $k$, whose circuits of rank $k-1$ are the subspaces in $\mathcal{S}$.
    \label{paving_construction}
\end{proposition}

Another crucial concept stemming from matroid theory is the duality of $q$-matroids.
This is again inspired by the usual matroid duality.
\begin{proposition}[{\cite[Theorem 42]{jp2018}}]
    Let $\M=(E,\rho)$ be a $q$-matroid and $\B$ its collection of bases. Let $\perp$ denote the orthogonal complement w.r.t.\ a non-degenerated bilinear form on $E$.
    Define the function $\rho^*$ by setting
    \begin{align*}
    	&\rho^*:\cL(E)\rightarrow\Z_{\geq 0}, \quad    	\rho^*(V)=\dim(V)+\rho(V^\perp)-\rho(E).
    \end{align*}
    
    Then $\M^*=(E,\rho^*)$ is a $q$-matroid, called the \textbf{dual $q$-matroid} of $\M$. Moreover, the collection of bases $\B^*$ of $\M^*$ are the orthogonal complements of the elements in $\B$.  
\end{proposition}
By definition, the dual rank is given by $\rho^*(\M^*)=n-\rho(\M)$ and we naturally have $(\M^*)^*=\M$. We want to emphasize here that there exists a bijection between $q$-matroids in $\F_q^n$ of rank $k$ and those of rank $(n-k)$ given by the map which sends a $q$-matroid to its dual, see \cite[Section 8]{jp2018}. Therefore we may restrict ourselves to considering $q$-matroids of rank $k$ with  $0\leq k\leq\lfloor\frac{n}{2}\rfloor$ where $n=\dim E$.

We conclude this subsection with an examples.

\begin{example}\label{exp:uniform}
	Set $E=\F_q^n$ for some prime power $q$ and integer $n\geq 1$.
    For a number $0\leq k\leq n$, we define the \textbf{uniform $q$-matroid} $\mathcal{U}_{k,n}(E):=(E,\rho)$ of rank $k$ and dimension $n$, where $\rho$ is given by
    \[
    	\rho(V)=\min\{k,\dim V\},\;\text{for all }V\in\cL(E).
    \]
    % So the collection of bases consists of all $\binom{n}{k}_q$ many $k$-dimensional subspaces of $E$.
    % Notice that  $\mathcal{U}_{k,n}(E)$ is a paving $q$-matroid, since the collection of circuits consist of all $\binom{n}{k+1}_q$ many $(k+1)$-spaces of $E$.
    % Moreover, $\mathcal{U}_{k,n}(E)$ arises from the construction in \Cref{paving_construction}, by setting $\mathcal{S}=\emptyset$.
    
    % Finally the dual $\mathcal{U}_{k,n}^*(E)$ is the uniform $q$-matroid $\mathcal{U}_{n-k,n}(E)$.
\end{example}

% \begin{example}
%     Let $E=\F_2^4$, $k=2$ and $\mathcal{S}$ be the collection of the $2$-dimensional subspaces $V,W$ generated by the rows of these two matrices $A$ and $B$, respectively:
%     \[
%     A=\begin{pmatrix}
%         1&0&0&0\\
%         0&1&0&0\\
%     \end{pmatrix},\quad
%     B=\begin{pmatrix}
%         0&0&1&0\\
%         0&0&0&1\\
%     \end{pmatrix}.
%     \]

%     We clearly have $\dim(V\cap W)=0$, and so the collection $\mathcal{S}$ satisfies the assumptions of \Cref{paving_construction}.
%     Thus this yields a paving $q$-matroid of rank $2$ in dimension $4$, having all $\binom{4}{2}_2=35$ many $2$-spaces as bases, except the two in $\mathcal{S}$.
%     \label{q_mat_example}
% \end{example}

\subsection{Rank-metric codes and representable \texorpdfstring{$q$}{q}-matroids}
\label{rank_metric_code}

This subsection serves as a brief introduction to algebraic coding theory, more specifically we recall the basics about rank-metric codes and their connection to $q$-matroids, see~\cite{ab2023_2,ab2023_1} for further details.

We start with rank-metric codes. For this purpose we endow the vector space $\F_{q^m}^n$ with the so called \textbf{rank distance metric}, defined as $d_{\text{rk}}(v,w):=\text{rk}(v-w)$ for every $v,w\in\F_{q^m}^n$, where for $v=(v_1,\ldots,v_n)\in\F_{q^m}^n$ we set $\text{rk}(v):=\dim_{\F_q}\langle v_1,\ldots,v_n\rangle_{\F_q}$.

\begin{definition}
    We call an $\F_{q^m}$-linear subspace $\C\leq\F_{q^m}^n$ a \textbf{rank-metric code} in $\F_{q^m}^n$. Its \textbf{minimal rank distance} is
    $$d_{\text{rk}}(\C):=\min\{\text{rk}(v)\;|\;v\in\C,v\not=0\}.$$
    
    Moreover if $\C$ has dimension $k$, we call a matrix $G\in\F_{q^m}^{k\times n}$ whose rows generate $\C$, a \textbf{generator matrix} of $\C$.
    Finally we denote by $\C^\perp$ the \textbf{dual code} of $\C$, which is the orthogonal complement of $\C$ with respect to the standard dot product given by $v\cdot w=\sum_{i=1}^n v_i w_i$ for all $v,w\in\F_{q^m}^n$.
\end{definition}

In the literature, sometimes the above definition of a rank-metric code refers to a \textbf{vector rank-metric code}, to distinguish it from the so-called \textit{linear matrix rank-metric code}. In our paper, the term rank-metric code will always refer to a vector rank-metric code, unless otherwise specified.
% A \textbf{matrix rank-metric code} is a subset $\C\subseteq\F_q^{n\times m}$, where $\F_q^{n\times m}$ is endowed with the metric $d(A,B):=\text{rk}(A-B)$ for all $A,B\in\F_q^{n\times m}$. It has \textbf{minimal rank-distance} given by $$d(\C):=\min\{\text{rk}(M-N)\;|\;M,N\in\C,M\not= N\}.$$
% If $\C$ is not only a subset of $\F_q^{n\times m}$ but also a $\F_q$-linear subspace, it is called \textbf{linear}.

% These vector and linear matrix rank-metric codes are related to each other, see \cite{gj2022,ab2023_1} for more details.
% In our paper, the term rank-metric code will always refer to a vector rank-metric code, unless otherwise specified.
 
In analogy with usual matroids, we now define representable $q$-matroids as those which arise from rank-metric codes.

\begin{proposition}[{\cite[Theorem 24]{jp2018}}]\label{rank_func_rep_q_mats}
    Let $\F_{q^m}$ be a field extension of $\F_q$ and let $G$ be a $k\times n$-matrix with entries in $\F_{q^m}$ where $1\le k \le n$.
    Assume that $G$ has rank $k$.
    Define a map $\rho:\cL(\F_q^n)\rightarrow\Z_{\geq 0}$ via
    $$\rho(V)=\text{rk}_{\F_{q^m}}(GY^T),$$
    where $Y$ is a matrix such that the subspace $V$ is the row space of $Y$. Then $\M_G:=(\F_q^n,\rho)$ is a $q$-matroid of rank $k$, called the \textbf{$q$-matroid represented by $G$}.
\end{proposition}


Note that we can view $G$ as the generator matrix of a rank-metric code $\C\leq\F_{q^m}^n$, therefore $\M_G$ is in the literature also sometimes named the \textbf{$q$-matroid associated to $\C$} and denoted by $\M_\C$, see for instance \cite{ab2023_2}.


\begin{definition}
    A $q$-matroid $\M$ is \textbf{representable} if there exist $m\geq 1$ and a rank-metric code $\C\leq\F_{q^m}^n$, such that the associated $q$-matroid $\M_\C$ equals $\M$. 
\end{definition}

These $q$-matroid representations behave well under duality:

\begin{proposition}\cite[Theorem 48]{jp2018}\label{Duality_rep_q_mats}
    Let $\M_\C$ be a representable $q$-matroid, with associated rank-metric code $\C\leq\F_{q^m}^n$. Then $\M_\C^*=\M_{\C^\perp}$. 
\end{proposition}

% Let us illustrate the above theory with an example.

% \begin{example}
%     Let $\M_G=(\F_2^4,\B)$ be the $q$-matroid of rank $2$ in dimension $4$ associated to the matrix
%     $$\begin{pmatrix}
%      1&\alpha&0&0\\0&0&1&\alpha^2
%      \end{pmatrix}\in\F_{2^4}^4,$$
%      where $\alpha$ is the primitive element of the field extension $\F_{2^4}$ over $\F_2$. Then one can compute that the collection of bases of $\M_G$ is given by
%      $$\B=\binom{\F_2^4}{2}_2\setminus\Bigg\{\begin{pmatrix}1&0&0&0\\0&1&0&0\end{pmatrix},\begin{pmatrix}0&0&1&0\\0&0&0&1\end{pmatrix}\Bigg\},$$
%      which coincides with the $q$-matroid from \Cref{q_mat_example}.
%      Therefore, this $q$-matroid is representable.
% \end{example}

% We end this subsection by describing a special kind of matrix rank-metric code, which we revisit in \Cref{lower_bound}.

We end this subsection by describing a special kind of vector rank-metric code, which we revisit in \Cref{lower_bound}.
The following result plays a key role in the definition of this special class of codes and was first proved by Delsarte~\cite{d1978}.

\begin{proposition}[Singleton-like Bound]\label{Singelton_blound}
    Let $\C\leq\F_{q^m}^{n}$ be a $k$-dimensional vector rank-metric code with minimal rank distance $d:=d_{\text{rk}}(\C)$. Then we have
    \[
         k\leq n-d+1.
	\]
\end{proposition}

% \begin{proposition}[Singleton-like Bound]\label{Singelton_blound}
%     Let $\C\leq\F_q^{n\times m}$ be a $k$-dimensional matrix rank-metric code with minimal rank distance $d:=d(\C)$. Then we have
%     \[
%         k\leq\max(m,n)(\min(m,n)-d+1).
% 	\]
% \end{proposition}

% Since vector rank-metric and linear matrix rank-metric codes are related, we can also state the above singleton-like bound for the former codes, see \cite[Remark 3.6.]{g2019}. More precisely if $\C\leq\F_{q^m}^{n}$ is a $k$-dimensional vector rank-metric code with minimal rank distance $d:=d_{\text{rk}}(\C)$, it must then hold that
% \[
%         k\leq n-d+1.
% \]

% Matrix or vector rank-metric codes attaining the above bounds are called \textbf{maximal (matrix/vector) rank-metric codes (MRD codes)}.  Their connection to $q$-matroids is stated in the next example.

Vector rank-metric codes attaining the above bound are called \textbf{maximal (vector) rank-metric codes (MRD codes)}.  Their connection to $q$-matroids is stated in the next example.

\begin{example}
    Let $0<k<n$ and $\C$ be a $k$-dimensional vector MRD code in $\F_{q^m}^n$. Such a code can only exist in the case $m\geq n$ and the $q$-matroid associated to $\C$ is the uniform $q$-matroid $\mathcal{U}_{k,n}(\F_q^n)$ described in \Cref{exp:uniform}, see~\cite[Example 2.4.]{gj2023}. 
    In other words, the uniform $q$-matroid $\mathcal{U}_{k,n}(\F_q^n)$ is representable over $\F_{q^m}^n$ if and only if $m\geq n$.
\end{example}


\section{A lower bound for the number of \texorpdfstring{$q$}{q}-matroids}
\label{lower_bound}

In this section, we give a lower bound on the number of $q$-matroids in a fixed dimension. For this purpose, we use the so-called subspace codes and relate them with the paving $q$-matroid construction presented in \Cref{q_mats}.

We start with a brief overview of subspaces codes, see \cite{skk2008,hk2017} for more details.
We consider the set $\cL(\F_q^n)$ endowed with the \textbf{subspace distance} $d_{\text{S}}$, defined as
\[
	d_{\text{S}}(V,W):=\dim(V)+\dim(W)-2\dim(V\cap W)\quad \text{for all } V,W\in\cL(\F_q^n).
\]
Then the pair $(\cL(\F_q^n),d_{\text{S}})$ is a metric space.

\begin{definition}
    A non-empty subset $\C\subseteq\cL(\F_q^n)$ is called a \textbf{subspace code}. The \textbf{minimum subspace distance} is given by
    $$d_\text{S}(\C):=\min\{d_\text{S}(V,W)\;|\;V,W\in\C,V\not=W\}.$$
    Let $\C$ be a subspace code. If the dimensions of all elements of $\C$ are equal, it is called a \textbf{constant dimension code (CDC)}.
    We denote by $A_q(n,d;k)$ the maximal cardinality of a constant dimension code $\C\subseteq\binom{\F_q^n}{k}_q$ with minimal subspace distance $d_\text{S}(\C)\geq d$.
\end{definition}

One part of our result relies on the following lower bound on the maximal cardinality of constant dimension codes.
\begin{proposition}\label{lower_bound_max_card_CDC}
For $2k\leq n$ and $d\geq 4$, it holds that
\[
    q^{(n-k)\cdot(k-\frac{d}{2}+1)}\leq A_q(n,d;k).
\]
\end{proposition}

% This inequality is a consequence of \cite[Section IV]{skk2008}.
% The authors essentially use a specific lifting construction to obtain constant dimension codes from MRD codes in $\F_q^{k\times (n-k)}$.
% This is achieved by putting an identity matrix as prefix in front of the codewords of the MRD code and then taking the row space of the resulting matrices.
% The resulting constant dimension code inherits the distance properties of its underlying matrix rank-metric code. Moreover it also preserves the cardinality of the underlying matrix rank-metric code, which finally yields the lower bound on $A_q(n,d;k)$.    

Next, we describe a connection between CDC's and paving $q$-matroids introduced in \Cref{q_mats}. The proof of the following lemma can be found in our paper \cite{dl2024}.

\begin{lemma} \label{connection_paving_cdc}
	Let $\mathcal{S}\subseteq\cL(\F_q^n)$ be a CDC of dimension $k$ and minimal distance at least $4$.
	Then $\mathcal{S}$ fulfills the assumption of \Cref{paving_construction} and therefore yields a paving $q$-matroid.
\end{lemma}

% \begin{proof}
% 	Let $V,W\in \mathcal{S}$ be two distinct subspaces.
% 	By construction we thus have
% 	\[
% 	d_{\mathcal{S}}(V,W)=\dim(V) +\dim(W)-2\dim(V\cap W)\ge 4.
% 	\]
% 	Since $V,W$ are both $k$-dimensional subspaces, this yields
% 	$
% 	\dim(V\cap W)\le k-2
% 	$
% 	as desired.
% \end{proof}

% Let us mention here another connection between paving $q$-matroids and graph theory. We will however not exploit that observation in this article.

% \begin{remark}\label{grassmann_graph_remark}
%     Consider the simple graph $J_q(n,k)$, with vertex set given by $\mathcal{V}=\binom{\F_q^n}{k}_q$ and the adjacency relation for two vertices $V,W\in\mathcal{V}$ defined via the condition $\dim(V\cap W)=k-1$. This graph is called the \textbf{Grassmann graph} or \textbf{$q$-Johnson graph}, see \cite{hk2017}.
    
%     An \textbf{independent set} $I$ of $J_q(n,k)$ is a subset of the vertex set such that no two vertices in this set are connected by an edge, i.e., a subset $I\subseteq\mathcal{V}$ such that for every two vertices $V,W\in I$ the inequality $\dim(V\cap W)\leq k-2$ is satisfied.
    
%     Thus, the independent sets of $J_q(n,k)$ yield paving $q$-matroids via \Cref{paving_construction}. 
%   	Note that this is the $q$-analogue of the well-known statement for usual matroids, namely that the independent sets of the Johnson graph are in one-to-one correspondence with sparse paving matroids.
% \end{remark}

Now we prove a lower bound on the number $\mathcal{N}_{q}(k,n)$ of $q$-matroids of rank $k$ in a fixed dimension $n\geq 1$, and subsequently give the lower bound on the number of all $q$-matroids, which will be an immediate consequence of \Cref{lower_bound_q_mats}. Recall that we can restrict ourselves to the case $0\leq k\leq \lfloor\frac{n}{2}\rfloor$ by $q$-matroid duality.

\begin{theorem}
    Let $n\geq 4$ be an integer and $2\leq k\leq \lfloor\frac{n}{2}\rfloor$. Then the number $\mathcal{N}_{q}(k,n)$ of $q$-matroids of rank $k$ on $\F_q^n$ satisfies
    \[
        2^{q^{(n-k)\cdot(k-1)}}< \mathcal{N}_{q}(k,n).
    \]
    \label{lower_bound_q_mats}
\end{theorem}

\begin{proof}
    Let $\mathcal{S}$ be a CDC of maximal size, having minimal subspace distance $d_\text{S}(\mathcal{S})=4$. Since $2 k\leq n$ all the conditions of the estimation in \Cref{lower_bound_max_card_CDC} are fulfilled and thus
    \[
        Q:=q^{(n-k)\cdot(k-4/2+1)}\leq |\mathcal{S}|.
    \]
    Moreover by \Cref{connection_paving_cdc} the set $\mathcal{S}$ forms a paving $q$-matroid in the sense of \Cref{paving_construction}. Then all its subsets also satisfy the intersection-dimension condition of \Cref{paving_construction}, therefore all of them form paving $q$-matroids as well. Furthermore all of these are indeed different, since they possess different circuit collections.~This implies that we have at least $2^Q$-many paving $q$-matroids of rank $k$ on $\F_q^n$, which yields the desired inequality and completes the proof.\end{proof}


\begin{corollary}[{\cite{dl2024}}]
    Let $n\geq 4$ and let $\mathcal{N}_{q}(n)$ denote the number of all $q$-matroids on $E=\F_q^n$. Then
    \[
        2^{q^{(n-\lfloor\frac{n}{2}\rfloor)\cdot(\lfloor\frac{n}{2}\rfloor-1)}}< \mathcal{N}_{q}(n).
    \]
    \label{corrollary_lower_bound_q_mats}
\end{corollary}

% \begin{proof}
%     This is just \Cref{lower_bound_q_mats} applied to the case $k=\lfloor\frac{n}{2}\rfloor$.  
% \end{proof}

% Let us note here that the above bound is not sharp and one way to improve it would be to sum up all bounds for $2\leq k\leq \lfloor\frac{n}{2}\rfloor$.  

% \section{An upper bound for the number of representable $q$-matroids}
\section{An upper bound for representable \texorpdfstring{$q$}{q}-matroids}
\label{upper_bound}


\subsection{Zero patterns and their connection to representable \texorpdfstring{$q$}{q}-matroids}

In this subsection, we start by giving a brief introduction to the theory of zero patterns, see \cite{rbg2000} for more details. Afterwards, we explain how these patterns are related to representable $q$-matroids. 

\begin{definition}
    We call a string of length $m$ over the alphabet $\{0,*\}$ a \textbf{zero pattern} of length $m$. Let $\K$ be field and $a\in\K$, then set
    \[
        \delta(a):=\begin{cases}
            \;\hfil0\;&\text{if } a=0,\\
            \;\hfil*\;&\text{otherwise}.
        \end{cases}
    \]
    When $a=(a_1,\ldots,a_s)\in\K^s$ we apply $\delta$ coordinate-wise, i.e., $\delta(a)=(\delta(a_1),\ldots,\delta(a_s))$ which is called the \textbf{zero pattern of $a$}.
    
    Lastly let $f=(f_1,\ldots,f_m)$ be a sequence of functions $f_i:D\rightarrow\K$ on a common domain set $D$. Now for an $a\in D$ we call $\delta(f,a):=(\delta(f_1(a)),\ldots,\delta(f_m(a)))$ a \textbf{zero pattern of $f$}.
\end{definition}

For our purposes, we only consider the case that $D=\K^s$ and the $f_i$ are polynomials in $K[x_1,\ldots,x_s]$. Moreover, let us denote the number of all zero patterns of $f$ as $a$ ranges over $D$ by $Z_\K(f)$.

We now state a result concerning an upper bound on the number $Z_\K(f)$, which we use later to bound the number of representable $q$-matroids for fixed rank. The proof of the following Theorem is a direct consequence of \cite[Theorem 1.3]{rbg2000}.

\begin{theorem}[{\cite{rbg2000}}]
    \label{zero_patterns_est}
    Let $f=(f_1,\ldots,f_m)$ be sequence of polynomials in $s$ variables over the field~$\K$, all having degree less or equal $d$ and assume $m\geq s$. Then for $d=1$ we have  $Z_\K(f)\leq\sum_{j=0}^s\binom{m}{j}$ and for $d\geq 2$ it holds that
    \[
        Z_\K(f)\leq\binom{md}{s}.
    \]
\end{theorem}

Next, we want to relate these zero patterns to representable $q$-matroids. To this end, we define the following sequence of polynomials.

\begin{definition}\label{Construction_polynomials}
    Let $n\geq 1$ be an integer and $1\leq k\leq n$. Moreover let us consider the vector space $\F_q^n$ and fix a total ordering on the set of all $k$-dimensional subspaces of $\F_q^n$, i.e., $U_1,\ldots,U_{\binom{n}{k}_q}$. We define the $(k\times n)$-matrix
    \[
        G=\begin{pmatrix}
            x_{1,1}&\cdots&x_{1,n}\\
            \vdots&\ddots&\vdots\\
            x_{k,1}&\cdots&x_{k,n}
        \end{pmatrix},
    \]
    where the $x_{i,j}$'s are the indeterminates of a polynomial ring $P$ over the algebraic closure of $\F_q$, i.e., $P=\overline{{\F_q}}[x_{i,j}\;|\;1\leq i \leq k, 1\leq j\leq n]$.
    Each $k$-dimensional space $U_i\in\binom{\F_q^n}{k}_q$ can be regarded as the row space of a matrix $Y_{U_i}\in\F_q^{k\times n}$, i.e., $\text{rowspan}_{\F_q}(Y_{U_i})=U_i$ for all $i=1,\ldots,\binom{n}{k}_q$.
    Now we define $\binom{n}{k}_q$-many homogeneous polynomials of degree $k$ in $P$, via
    \[
        f_{U_i}(\boldsymbol{x})=\det(G\cdot Y_{U_i}^T)\quad \text{for all } i=1,\ldots,\binom{n}{k}_q,
    \]
    where we set $\boldsymbol{x}=(x_{i,j})_{1\leq i\leq k,1\leq j\leq n}$.\ Finally denote by $\mathcal{F}_{n,k}$ the sequence of the above polynomials, i.e., $\mathcal{F}_{n,k}:=(f_{U_i})_{1\leq i\leq\binom{n}{k}_q}$.
\end{definition}

The following lemma provides a characterization of representable $q$-matroids in terms of zero patterns of the sequence of polynomials $\mathcal{F}_{n,k}$. Let us denote the set of all zero patterns of $\mathcal{F}_{n,k}$ by $P_{\overline{{\F_q}}}(\mathcal{F}_{n,k})$.

\begin{lemma}
\label{zero_patterns_rep_q_mat_}
    Let $\M$ be a $q$-matroid of rank $k$ on $\F_q^n$ and $\B$, $\mathcal{NB}$ its collection of bases and non-bases, respectively. Then $\M$ is representable if and only if there exists a zero pattern of $\mathcal{F}_{n,k}$ for some $u\in\overline{{\F_q}}^{kn}$ of the form
    \[
        \delta(\mathcal{F}_{n,k},u)=(f_{U_i}(u))_{1\leq i\leq\binom{n}{k}_q}=
        \begin{cases}
            \;\hfil0\;&\emph{if } U_i\in\mathcal{NB},\\
            \;\hfil*\;&\emph{if } U_i\in\B.
        \end{cases}
    \]
\end{lemma}

\begin{proof}
    On the one hand, if $\M$ is representable, then there exists a $(k\times n)$-matrix $G$ in $\F_{q^m}^{k\times n}$, for some $m\geq 1$, representing $\M$, and the entries of $G$ from a vector $u\in\overline{{\F_q}}^{kn}$ such that all $f_{U_i}$ corresponding to non-bases vanish and those corresponding to bases do not. Thus we get the above described zero pattern in $P_{\overline{{\F_q}}}(\mathcal{F}_{n,k})$, which concludes the proof of the first direction.
    
    On the other hand, given such a zero pattern $\delta(\mathcal{F}_{n,k},u)$ for some $u\in\overline{{\F_q}}^{kn}$, the entries of $u$ form a full-rank $(k\times n)$-matrix $G$ in some $\F_{q^m}^{k\times n}$. Therefore, $\M$ is represented by $G$ and thus representable, which concludes the proof of the second direction.   
\end{proof}

Now denote by $\mathcal{R}_{q}(k,n)$ the number of representable $q$-matroids on $\F_q^n$ of rank $k$ in a fixed dimension $n\geq 1$. Then the following inequality is a direct consequence of \Cref{zero_patterns_rep_q_mat_}.

\begin{corollary}
    \label{num_rep_q_mats_num_zero_patterns_ineq.}
    Let $0\leq k\leq n$. Then 
    \[
      \mathcal{R}_{q}(k,n)<Z_{\overline{{\F_q}}}(\mathcal{F}_{n,k}).
    \]
\end{corollary}

% \begin{proof}
%     By \Cref{zero_patterns_rep_q_mat_} every representable $q$-matroid has an associated zero pattern in $P_{\overline{{\F_q}}}(\mathcal{F}_{n,k})$. But there are zero patterns in $P_{\overline{{\F_q}}}(\mathcal{F}_{n,k})$ which do not yield $q$-matroids of rank $k$, for instance $(0,\ldots,0)$ if $k\not=0$. Thus, we obtain the strict inequality stated above.
% \end{proof}

% \begin{remark}
%     The above lemma can be used to algorithmically test, whether a given $q$-matroid is representable over some field using Gr\"obner bases.
%     This parallels the well-known algorithm to test matroid representability over algebraically closed fields for instance described in~\cite[Theorem~6.8.14]{o2011}.
% \end{remark}


\subsection{The upper bound and the proof of the main result}

In \Cref{lower_bound} we established a lower bound on the number of all $q$-matroids. In this subsection, we turn to the discussion of an upper bound on the number of representable $q$-matroids. This provides us with the last piece for the discussion about the asymptotic behavior of the representable $q$-matroids. We conclude the subsection with the proof of our main result, \Cref{q_mat_asymptotics}.

As in \Cref{lower_bound} we first prove an upper bound on the number $\mathcal{R}_{q}(k,n)$ for fixed $k$ and $n$ and afterward give the upper bound on the number of all representable $q$-matroids, which will be an immediate consequence of \Cref{upper_bound_rep_q_mats}.

Recall again that we restrict ourselves to the case $0\leq k\leq\lfloor\frac{n}{2}\rfloor$ by $q$-matroid duality.

\begin{theorem}
    Let $n\geq 2$ and $1\leq k\leq\lfloor\frac{n}{2}\rfloor$. Then the number of representable $q$-matroids of rank $k$ satisfies the following upper bounds, depending on the rank $k$.
    \begin{enumerate}
        \item If $k=1$ it holds that
        \[
            \mathcal{R}_{q}(1,n)<q^{\log_q(n)+n^2+n\log_q(e)}+1.
        \]

        \item If $2\leq k\leq \lfloor\frac{n}{2}\rfloor$ it holds that
        \[
            \mathcal{R}_{q}(k,n)< \Big(\frac{111}{32}\Big)^{kn}q^{k^2n^2-k^3n+k\log_q(e)n}=q^{n^2k^2-nk^3+nk\log_q(e)+nk\log_q(\frac{111}{32})}.
        \]

        \item Moreover there exists a bound independent of $k$, which holds for every $k\in\{2,\ldots,\lfloor\frac{n}{2}\rfloor\}$. This bound is given by
        \[
            \mathcal{R}_{q}(k,n)< \Big(\frac{111}{32}\Big)^\frac{n^2}{2}q^{\frac{n^2}{4}+\log_q(e)\frac{n^2}{2}}=q^{\frac{n^2}{4}+\log_q(e)\frac{n^2}{2}+\log_q(\frac{111}{32})\frac{n^2}{2}}.
        \]
    \end{enumerate}
    \label{upper_bound_rep_q_mats}
\end{theorem}

% We want to give a short remark concerning Part 1 of \Cref{upper_bound_rep_q_mats}.

% \begin{remark}
%     We know that all different $q$-matroids of rank $1$ on $\F_q^n$ are determined by selecting the dimension of their loop spaces ranging from $0$ to $n-1$. Moreover, all these $q$-matroids are representable. The reason for this is as follows. Firstly all $q$-matroids of rank $1$ having the same loop space dimension are isomorphic. So without loss of generality, we can describe a loop space of dimension $0\leq k\leq n-1$ by the form
%     \[
%         \begin{cases}
%             \;\hfil\langle e_1,\ldots,e_k\rangle\;&\text{if } k\not=0,\\
%             \;\hfil\{0\}\;&\text{otherwise}.
%         \end{cases}
%     \]
%     Secondly given a $q$-matroid of rank $1$ having a loop space of dimension $k$, we can construct its representing matrix $G$ via
%     \[
%         \begin{cases}
%             \;\hfil G=\begin{pmatrix}0,\ldots,0,1,\alpha,\ldots,\alpha^{n-k-1}\end{pmatrix}\in\F_{q^k}^{1\times n}\;&\text{if } k\not=0 \text{ and}\\
%             \;\hfil G=\begin{pmatrix}1,\beta,\ldots,\beta^{n-1}\end{pmatrix}\in\F_{q^n}^{1\times n}\;&\text{otherwise},
%         \end{cases}
%     \]
%     where $\alpha,\beta$ are primitive elements of the field extensions $\F_{q^k}/\F_q$ and $\F_{q^n}/\F_q$, respectively.
%     Therefore we can compute the exact number $\mathcal{R}_{q}(1,n)$ by the formula
%     \[
%         \mathcal{R}_{q}(1,n)=\sum_{i=0}^{n-1}\binom{n}{i}_q.
%     \]
%     Thus the estimation in \Cref{upper_bound_rep_q_mats} Part 1 is strictly larger than this sum. However, the estimation is more than enough for our purposes.
% \end{remark}

To prove \Cref{upper_bound_rep_q_mats} we first need the following lemma. %The proof can be found in our paper \cite{dl2024}.

\begin{lemma}[{\cite{dl2024}}]
    \label{q_binomial_inequality}
    Let $n\geq 2$ and $1\leq k\leq\lfloor\frac{n}{2}\rfloor$. Then the following inequality holds
    \[
        nk\leq\binom{n}{k}_q.
    \]
\end{lemma}

% \begin{proof}
%     By \Cref{Estimations_q_binom} we know that $\binom{n}{k}_q$ can be lower bounded by $q^{(n-k)k}$. Using that one computes
%     \[
%         \binom{n}{k}_q\geq q^{(n-k)k}\geq q^{\lceil\frac{n}{2}\rceil k}\geq q^{\lceil\frac{n}{2}\rceil+k}\geq q^{\lceil\frac{n}{2}\rceil}k\geq nk,
%     \]
%     where for the last inequality we used that $q^{\lceil\frac{n}{2}\rceil}\geq n$ for all $n\geq 2$, which finishes the proof.
% \end{proof}

Now we are ready to prove \Cref{upper_bound_rep_q_mats}, using the theory of zero patterns of polynomials.
\begin{proof}[Proof of \Cref{upper_bound_rep_q_mats}]
    Our goal is to apply the inequalities of \Cref{zero_patterns_est} to the sequence of polynomials $\mathcal{F}_{n,k}$ from \Cref{Construction_polynomials} as we know from \Cref{num_rep_q_mats_num_zero_patterns_ineq.} that $\mathcal{R}_{q}(k,n)<Z_{\overline{{\F_q}}}(\mathcal{F}_{n,k})$ holds. 
    \Cref{q_binomial_inequality} ensures that the conditions specified in \Cref{zero_patterns_est} hold for the parameters $m=\binom{n}{k}_q$, $s=kn$, and $d=k$.
    
    For the first statement, keeping in mind that $k=1$, we use the first inequality in \Cref{zero_patterns_est}, which then gives us
    \[
    \mathcal{R}_{q}(1,n)< Z_{\overline{{\F_q}}}(\mathcal{F}_{n,1})\leq1+\sum_{i=1}^n\binom{\binom{n}{ 1}_q}{i}.
    \]
    This sum can be upper bounded by
    \[
        \sum_{i=1}^n\binom{\binom{n}{ 1}_q}{i}<\sum_{i=1}^n\Bigg({\frac{q^ne}{i}}\Bigg)^i
        < n(q^ne)^n
        = nq^{n^2}e^n
        = q^{\log_q(n)+n^2+n\log_q(e)},
    \]
    
    % \begin{align*}
    %     \sum_{i=1}^n\binom{\binom{n}{ 1}_q}{i}&<\sum_{i=1}^n\Bigg({\frac{q^ne}{i}}\Bigg)^i\\
    %     & < n(q^ne)^n\\
    %     &= nq^{n^2}e^n\\
    %     &= q^{\log_q(n)+n^2+n\log_q(e)},
    % \end{align*}
    where the first inequality follows from the general estimate $\binom{n}{1}_q\leq q^{n}$, as well as the general bound $\binom{n}{k}<(\frac{ne}{k})^k$ for the binomial coefficient. In total, this proves the first statement.
    
    Now to prove the second statement of \Cref{upper_bound_rep_q_mats} we consider $2\leq k\leq \lfloor\frac{n}{2}\rfloor$ and therefore  we use the second inequality from \Cref{zero_patterns_est}, which then yields 
    \[
        \mathcal{R}_{q}(k,n)< Z_{\overline{{\F_q}}}(f)\leq{{\binom{n}{k}_q k}\choose kn}.
    \]
    This binomial coefficient can be upper bounded by
    \begin{align*}
        {{\binom{n}{k}_q k}\choose kn}&<\Bigg({\frac{\frac{111}{32}q^{(n-k)k}e}{n}}\Bigg)^{kn} < \Big(\frac{111}{32}\Big)^{kn}q^{n^2k^2-nk^3}e^{kn}= q^{n^2k^2-nk^3+nk\log_q(e)+nk\log_q(111/32)},
    \end{align*}
    
    % \begin{align*}
    %     {{\binom{n}{k}_q k}\choose kn}&<\Bigg({\frac{\frac{111}{32}q^{(n-k)k}e}{n}}\Bigg)^{kn}\\
    %     & <\Bigg(\frac{111}{32}q^{(n-k)k}e\Bigg)^{kn}\\
    %     &= \Big(\frac{111}{32}\Big)^{kn}q^{n^2k^2-nk^3}e^{kn}\\
    %     &=\Big(\frac{111}{32}\Big)^{kn}q^{n^2k^2-nk^3+nk\log_q(e)}\\
    %     &= q^{n^2k^2-nk^3+nk\log_q(e)+nk\log_q(111/32)},
    % \end{align*}
    where the first inequality follows from the bound for the $q$-binomial coefficient from \Cref{Estimations_q_binom} and the general bound on the binomial coefficient from the previous case. This however proves the second statement. For the third statement, one can use the fact that $n^2k^2-nk^3+nk\log_q(e)+nk\log_2(111/32)$ is strictly increasing as a polynomial in $k$ over the interval $2\leq k\leq \frac{n}{2}$. Thus it attains its maximum in $k=\frac{n}{2}$ and so does $q^{n^2k^2-nk^3+nk\log_q(e)+nk\log_2(111/32)}$, which completes the proof of the third statement of \Cref{upper_bound_rep_q_mats}.  
\end{proof}

The desired upper bound on the number of representable $q$-matroids in $\F_q^n$, is now a direct consequence of \Cref{upper_bound_rep_q_mats}.

\begin{corollary}
    Let $n\geq 2$ be an integer and let $\mathcal{R}_{q}(n)$ be the number of all representable $q$-matroids on $E=\F_q^n$. Then we have
    \[
        \mathcal{R}_{q}(n)< 2\Big(q^{\frac{n^2}{4}+\log_q(e)\frac{n^2}{2}+\log_q(\frac{111}{32})\frac{n^2}{2}+\log_q(\frac{n}{2})}+q^{\log_q(n)+n^2+n\log_q(e)}+2\Big).
    \]
    
    % \begin{align*}
    %     \mathcal{R}_{q}(n)&< 2\Big(\frac{n}{2}\cdot q^{\frac{n^2}{4}+\log_q(e)\frac{n^2}{2}+\log_q(\frac{111}{32})\frac{n^2}{2}}+q^{\log_q(n)+n^2+n\log_q(e)}+2\Big) \notag\\
    %     &= 2\Big(q^{\frac{n^2}{4}+\log_q(e)\frac{n^2}{2}+\log_q(\frac{111}{32})\frac{n^2}{2}+\log_q(\frac{n}{2})}+q^{\log_q(n)+n^2+n\log_q(e)}+2\Big).
    % \end{align*}
    \label{corollary_upper_bound_rep_q_mats}
\end{corollary}

% \begin{proof}
%     By \Cref{Duality_rep_q_mats} we know that $\mathcal{R}_{q}(k,n)=\mathcal{R}_{q}(n-k,n)$, therefore we only need to consider $0\leq k\leq \lfloor\frac{n}{2}\rfloor$ and double our estimates.

%     The case $k=0$ yields exactly one representable $q$-matroid namely the trivial one, i.e., $\mathcal{U}_{0,n}(E)$. For the case $k=1$ we use \Cref{upper_bound_rep_q_mats} (1) and for each of the cases $2\leq k\leq \lfloor\frac{n}{2}\rfloor$ we use \Cref{upper_bound_rep_q_mats} (3). This provides us a factor of $\lfloor\frac{n}{2}\rfloor-1$, that we bound by $\frac{n}{2}$. Altogether this yields the desired inequality and completes the proof.
% \end{proof}

With all preparations done, we are now finally able to prove \Cref{q_mat_asymptotics}.

\begin{proof}[Proof of \Cref{q_mat_asymptotics}]
Consider the ratio $\frac{\mathcal{R}_{q}(n)}{\mathcal{N}_{q}(n)}$.
\Cref{corrollary_lower_bound_q_mats} implies that the denominator of this fraction grows at least doubly exponentially while the numerator grows at most exponentially by~\Cref{corollary_upper_bound_rep_q_mats} when $n$ grows.
Thus in the limit this yields
\[
    \lim_{n\rightarrow\infty}\frac{\mathcal{R}_{q}(n)}{\mathcal{N}_{q}(n)}= 0.\qedhere
\]
\end{proof}


% Here is a citation~\cite{greenwade93} with URL. Here is another
% citation~\cite{MR4}, which is the earliest-numbered Math Review with a
% functioning DOI: MR0000004. Finally, a reference to
% equation~\eqref{eqn:eq1}. And a reference to \Cref{fig:plot}.

\acknowledgements{We thank Gianira Nicoletta Alfarano and Charlene Weiß for fruitful discussions. We are grateful to Kai-Uwe Schmidt for introducing us to the topic of $q$-matroids.}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography



\end{document}
