\section{Introduction}
``\emph{Sometimes a picture is worth the proverbial thousand words; sometimes a few well-chosen words are far more effective than a picture}'' -- \cite{feiner1991automating}.
Modeling how visual and linguistic information can jointly contribute to coherent and effective communication is a longstanding open problem with implications across cognitive science.  As \cite{feiner1991automating} already observe, it is particularly important for automating the understanding and generation of text--image presentations.\\
Theoretical models have suggested that images and text fit together into integrated presentations via \emph{coherence relations} that are analogous to those that connect text spans in discourse; see  \cite{8397019} and Section~\ref{sec:related}.
This paper follows up this theoretical perspective through systematic corpus investigation.
We are inspired by research on text discourse, which has led to large-scale corpora with information about discourse structure and discourse semantics. The Penn Discourse Treebank (PDTB) is one of the most well-known examples \cite{MiltsakakiPJW04,DBLP:conf/lrec/PrasadDLMRJW08}.  However, although multimodal corpora increasingly include discourse relations between linguistic and non-linguistic contributions, particularly for utterances and other events in dialogue \cite{cuayahuitl2015strategic,hunter:etal:2015},
to date there has existed no dataset describing the coherence of text--image presentations.  In this paper, we describe the construction of an annotated corpus that fills this gap, and report initial analyses of the communicative inferences that connect text and accompanying images in this corpus.
As we describe in Section \ref{img--txt}, our approach asks annotators to identify the presence of specific inferences linking text and images, rather than to use a taxonomy of coherence relations.  This enables us to deal with the distinctive discourse contributions of photographic imagery.  We describe our data collection process in Section~\ref{data}, showing that our annotation scheme allows us to get reliable labels by crowdsourcing.  We present analyses in Section~\ref{analysis} that show that our annotation highlights a range of cases where text and images work together in distinctive and theoretically challenging ways, and discuss the implications of our work for the understanding and generation of multimodal documents.  We conclude in Section \ref{conclusion} with a number of problems for future research.

\section{Introduction}
``\emph{Sometimes a picture is worth the proverbial thousand words; sometimes a few well-chosen words are far more effective than a picture}'' --\cite{feiner1991automating}.
Modeling how visual and linguistic information can jointly contribute to coherent and effective communication is a longstanding open problem with implications across cognitive science.  As \cite{feiner1991automating} already observe, it is particularly important for automating the understanding and generation of text--image presentations.\\
Theoretical models have suggested that images and text fit together into integrated presentations via \emph{coherence relations} that are analogous to those that connect text spans in discourse; see  \cite{8397019} and Section~\ref{sec:related}. \\
This paper follows up this theoretical perspective through systematic corpus investigation.

We are inspired by research on text discourse, which has led to large-scale corpora with information about discourse structure and discourse semantics. The Penn Discourse Treebank (PDTB) is one of the most well-known examples \cite{MiltsakakiPJW04,DBLP:conf/lrec/PrasadDLMRJW08}. However, although multimodal corpora increasingly include discourse relations between linguistic and non-linguistic contributions, particularly for utterances and other events in dialogue \cite{cuayahuitl2015strategic,hunter:etal:2015}, to date there has existed no dataset describing the coherence of text--image presentations.  In this paper, we describe the construction of an annotated corpus that fills this gap, and report initial analyses of the communicative inferences that connect text and accompanying images in this corpus.

As we describe in Section \ref{img--txt}, our approach asks annotators to identify the presence of specific inferences linking text and images, rather than to use a taxonomy of coherence relations.  This enables us to deal with the distinctive discourse contributions of photographic imagery.  We describe our data collection process in Section~\ref{data}, showing that our annotation scheme allows us to get reliable labels by crowdsourcing.  We present analyses in Section~\ref{analysis} that show that our annotation highlights a range of cases where text and images work together in distinctive and theoretically challenging ways, and discuss the implications of our work for the understanding and generation of multimodal documents.  We conclude in Section \ref{conclusion} with a number of problems for future research.

\vspace{3mm}
\section{Discourse Coherence and Text--Image Presentations}
\label{img--txt}
\label{sec:related}

We begin with an example to motivate our approach and clarify its relationship to previous work.
Figure~\ref{fig:example} shows two steps in an online recipe for a ravioli casserole from the RecipeQA data set \cite{DBLP:conf/emnlp/YagciogluEEI18}.
The image of Figure~\ref{fig:example}a shows a moment towards the end of carrying out the ``covering'' action of the accompanying text; that of Figure~\ref{fig:example}b shows one instance of the result of the ``spooning'' actions of the text.
\begin{figure*}[h!]
    \centering
    \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=4.7cm]{step2.png}
        \caption{\textsc{Text:} Cover with a single layer of ravioli.}
    \end{subfigure}%
    ~
    \begin{subfigure}[t]{0.5\columnwidth}
        \centering
        \includegraphics[width=4.7cm]{step7.png}
        \caption{\textsc{Text:} Let cool 5 minutes before spooning onto individual plates.}
    \end{subfigure}

    \caption{Two steps in a recipe from \protect{\cite{DBLP:conf/emnlp/YagciogluEEI18}} illustrating diverse inferential relationships between text and accompanying imagery in instructions. The recipe is from Autodesk Inc. www.instructables.com and is contributed by www.RealSimple.com.}
    \label{fig:example}

\end{figure*}
Cognitive scientists have argued that such images are much like text contributions in the way their interpretation connects to the broader discourse.
In particular, inferences analogous to those used to interpret text seem to be necessary with such images to recognize their spatio-temporal perspective \cite{cumming2017conventions}, the objects they depict \cite{abusch2012applying}, and their place in the arc of narrative progression \cite{mccloud1993understanding,cohn2013visual}.
In fact, such inferences seem to be a general feature of multimodal communication, applying also in the coherent relationships of utterance to co-speech gesture \cite{lascarides2009discourse} or the coherent relationships of elements in diagrams \cite{alikhani2018arrows,inproceedings}.

In empirical analyses of text corpora, researchers in projects such as the Penn Discourse Treebank \cite{MiltsakakiPJW04,DBLP:conf/lrec/PrasadDLMRJW08} have been successful at documenting such effects by annotating discourse structure and discourse semantics via coherence relations.
We would like to apply a similar strategy to text--image documents like that shown in Figure~\ref{fig:example}.
However, existing discourse annotation guidelines depend on the distinctive ways that coherence is signaled in text.
In text, we find syntactic devices such as structural parallelism, semantic devices such as negation, and pragmatic elements such as discourse connectives, all of which can help annotators to recognize coherence relations in text.
Images lack such features.
At the same time, characterizing the communicative role of imagery, particularly photographic imagery, involves a special problem: distinguishing the content that the author specifically aimed to depict from merely incidental details that happen to appear in the scene \cite{stone2015meaning}.

Thus, rather than start from a taxonomy of discourse relations like that used in PDTB, we characterize the different kinds of inferential relationships involved in interpreting imagery separately.
\begin{itemize}
\itemsep0em
\parsep0em
\parskip0em
    \item To characterize temporal relationships between imagery and text, we ask if the image gives information about the preparation, execution or results of the accompanying step.
    \item To characterize the logical relationship of imagery to text, we ask if the image shows one of several actions described in the text, and if it depicts an action that needs to be repeated.
    \item To characterize the significance of incidental detail, we ask a range of further questions (some relevant specifically to our domain of instructions), asking about what the image depicts from the text, what it leaves out from the text, and what it adds to the text.
\end{itemize}
Our approach is designed to elicit judgments that crowd workers can provide quickly and reliably.

This approach allows us to highlight a number of common patterns that we can think of as prototypical coherence relations between images and text.
Figure \ref{fig:example}a, for example, instantiates a natural \textbf{Depiction} relation: the image shows the action described in the text in progress; the mechanics of the action are fully visible in the image, but the significant details in the imagery are all reported in the text as well.
Our approach also lets us recognize more sophisticated inferential relationships, like the fact that Figure~\ref{fig:example}b shows an \textbf{Example:Result} of the accompanying instruction.
Many of the relationships that emerge from our annotation effort involve newly-identified features of text--image presentations that deserve further investigation: particularly, the use of loosely-related imagery to provide background and motivation for a multimodal presentation as a whole, and depictions of action that seem simultaneously to give key information about the context, manner and result of an action.

\vspace{5mm}
\section{Annotation Effort}
\label{data}

Work on text has found that text genre heavily influences both the kinds of discourse relations one finds in a corpus and the way those relations are signalled \cite{webber2009genre}.
Since our focus is on developing methodology for consistent annotation, we therefore choose to work within a single genre. We selected instructional text because of its concrete, practical subject matter and because of its step-by-step organization, which makes it possible to automatically group together short segments of related text and imagery.

\paragraph{Text--Image Pairs.} We base our data collection on an existing instructional dataset, RecipeQA~\cite{DBLP:conf/emnlp/YagciogluEEI18}. This is the only publicly available large-scale dataset of multimodal instructions. It consists of multimodal recipes---textual instructions accompanied by one or more images.

We excluded documents that either have multiple steps without images or that have multiple images per set. This was so that we could more easily study the direct relationship between an image and the associated text.
There are 1,690 documents with this characteristic in the RecipeQA train set.  To avoid overwhelming crowd workers, we further filtered those to retain only recipes with 70 or fewer words per step, for a final count of 516 documents (2,047 image--text pairs).

\paragraph{Protocol.}  We recruit participants through Amazon Mechanical Turk. All subjects were US citizens, agreed to a consent form approved by Rutgers's institutional review board, and were compensated at an estimated rate of USD 15 an hour.

\paragraph{Experiment Interface.} Given an image and the corresponding textual instruction from the dataset, participants were requested to answer the following 10 questions.

For Question 1, participants were asked to highlight the relevant part of the text. For the others, we solicited True/False responses.
\begin{enumerate}
\itemsep0em
\parsep0em
\parskip0em
    \item Highlight the part of the text that is most related to the image.
    \item The image gives visual information about the step described in the text.
    \item You need to see the image in order to be able to carry out the step properly.
    \item The text provides specific quantities (amounts, measurements, etc.) that you would not know just by looking at the picture.
    \item The image shows a tool used in the step but \emph{not} mentioned in the text.
    \item The image shows how to prepare before carrying out the step.
    \item The image shows the results of the action that is described in the text.
    \item The image depicts an action in progress that is described in the text.
    \item The text describes several different actions but the image only depicts one.
    \item One would have to repeat the action shown in the image many times in order to complete this step.
\end{enumerate}

The interface is designed such that if the answer to Question 8 is \tag{True}, the subject will be prompted with Question 9 and 10. Otherwise, Question 8 is the last question in the list.\footnote{The dataset and the code for the machine learning experiments are available at https://github.com/malihealikhani/CITE}

\paragraph{Agreement.} To assess the inter-rater agreement, we determine Cohen's $\kappa$ and Fleiss's  $\kappa$ values. For Cohen's $\kappa$, we randomly selected 150 image--text pairs and assigned each to two participants, obtaining a Cohen's $\kappa$ of 0.844, which indicates almost perfect agreement. For Fleiss's $\kappa$ \cite{fleiss1973equivalence,cocos2015effectively,banerjee1999beyond}, we randomly selected 50 text--image pairs, assigned them to five subjects, and computed the average $\kappa$. We obtain a score of 0.736, which indicates substantial agreement \cite{viera2005understanding}.

\section{Analysis}
\label{analysis}

\paragraph{Overall Statistics.}  Table~\ref{raw} shows the rates of true answers for questions Q2--Q10.

Subjects reported that in 17\% of cases the images did not give any information about the step described in the accompanying text.  Such images deserve further investigation to characterize their interpretive relationship to the document as a whole.  Our anecdotal experience is that such images sometimes provide context for the recipe, which may suggest that imagery, like real-world events \cite{hunter:etal:2015},  creates more flexible discourse structures than linguistic segments on their own.
\begin{table}[h!]
\centering
\begin{tabularx}{0.81\linewidth}{p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|}
\cline{2-10}
                            & Q2    & Q3    & Q4    & Q5    & Q6    & Q7    & Q8    & Q9    & Q10   \\ \hline
\multicolumn{1}{|l|}{True}  & 0.829 & 0.058 & 0.211 & 0.131 & 0.056 & 0.491 & 0.209 & 0.289 & 0.133 \\ \hline

\end{tabularx}
\vspace{3mm}
\caption{Rate of true answers for annotation questions Q2--Q10 across the corpus.}
\label{raw}
\end{table}

\begin{table}[h]
\centering
\begin{tabularx}{0.89\linewidth}{p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|p{9mm}|}
\cline{2-11}

    & Q1   & Q2**   & Q3**   & Q4**   & Q5   & Q6**   & Q7**   & Q8**  & Q9*   & Q10**  \\ \hline
\multicolumn{1}{|l|}{F1}  & 0.74 & 0.86 & 0.76 & 0.85 & 0.88 & 0.92 & 0.64 & 0.83 & 0.77 & 0.92 \\ \hline
\end{tabularx}
\vspace{4mm}

\caption{SVM classification accuracy: bag-of-words features; 80-20 train-test split; 5-fold cross validation. For the first question, this distinguishes highlighted text vs.\ its complement (excluded vs.\ included).  For the rest of the questions, this distinguishes text of true instances from text of false instances, and is different from majority class baseline $^*$ at $p < 0.04$, $t= -3.5$ and $^{**}$ at $p < 0.01$, $t>|2.49|$.}

\label{svm}
\end{table}

Subjects reported that the image was required in order to carry out the instruction only for 6\% of cases.
This suggests that subjects construe imagery as backgrounded or peripheral to the document, much as speakers regard co-speech iconic gesture as peripheral to speech \cite{schlenker2017gestural}.  Note, by contrast, that subjects characterized 12.7\% of images as introducing a new tool: this includes many cases where the same subjects say the image is not required.  In other words, subjects' intuitions suggest that coherent imagery typically does not contribute instruction content, but rather serves as a visual signal that facilitates inferences that have to be made to carry out the instruction regardless.  Our annotated examples, where imagery is linked to specific kinds of inferences, provide materials to test this idea.

\begin{figure}[h!]
\captionsetup[subfigure]{labelformat=empty}
    \centering
    \begin{subfigure}[t]{0.99\columnwidth}
        \centering
        \includegraphics[width=4.7cm]{step5-temporal.png}
        \caption{\textsc{Text:} Top with another layer of ravioli and the remaining sauce not all the ravioli may be needed. Sprinkle with the Parmesan.}
        \vspace{-2mm}
    \end{subfigure}%
    \caption{The image depicts both the action and the result of the action. The recipe is from Autodesk Inc.\ www.instructables.com and was contributed by www.RealSimple.com.}
    \label{fig:complex}
    \vspace{-4mm}
\end{figure}

\paragraph{The Complex Coherence of Imagery.}  Our annotation reveals cases where a single image does include more information than could be packaged into a single textual discourse unit (the proverbial thousand words).   In particular, such imagery participates in more complex coherence relationships than we find between text segments.  Multiple temporal relationships show this most clearly: 12\% of images that have any temporal relation have more than one.  For example, many images depict the action that is described in the text, while also showing preparations that have already been made by displaying the scene in which the action is performed.  Figure \ref{fig:complex} depicts the action and the result of the action. It also shows how to prepare before carrying out the action. Other images show an action in progress but nearing completion and thereby depict the result. For instance, the image that accompanies ``mix well until blended'' can show both late-stage mixing and the blended result. Looking at a few such cases closely, the circumstances and composition of the photos seem staged to invite such overlapping inferences.

Such cases testify to the richness of multimodal discourse, and help to justify our research methodology. The True/False questions characterize the relevant features of interpretation without necessarily mapping to single discourse relations.  For instance, Q4 and Q5 indicate inferences in line with an \textbf{Elaboration} relation; Q9 and Q10 indicate inferences in line with an \textbf{Exemplification} relation, as information presented in images show just one case of a generalization presented in accompanying text.  However, our data shows that these inferences can be combined in productive ways, in keeping with the potentially complex relevant content of images.
\begin{table*}[h]
\centering
\begin{tabularx}{0.5\linewidth}{|X|X|X|X|}
\hline
\multicolumn{4}{|>{\centering\arraybackslash}m{60mm}|}{Q4. Text has quantities not in image}                 \\ \hline
\multicolumn{2}{|c|}{True} & \multicolumn{2}{c|}{False} \\ \hline
1            & -4.1     & add        & -4.5      \\ \hline
cup          & -4.4     & place      & -4.9      \\ \hline
minutes      & -4.7     & put        & -5.0     \\ \hline
2            &  -4.7      & make       & -5.1      \\ \hline
1/2          & -4.9     & mix        & -5.1  \\\hline
\multicolumn{4}{c}{}  \\ \hline
\multicolumn{4}{|>{\centering\arraybackslash}m{60mm}|}{Q8. Image depicts action in progress}             \\ \hline
\multicolumn{2}{|c|}{True} & \multicolumn{2}{c|}{False} \\ \hline
add          &   -5.0       & 1         & -4.6     \\ \hline
mix       & -5.2     & cup           & -4.7     \\ \hline
place        & -5.3     & minutes         & -4.9     \\ \hline
bread        & -5.5     & 160     & -5.1     \\ \hline
make         &  -5.6        & put         & -5.2     \\ \hline
\end{tabularx}
\caption{Top five features of Multimodal Naive Bayes classifier for two classification problems and their corresponding log--probability estimates.}
\label{features}
\end{table*}
\paragraph{Information across modalities.}  We carried out machine learning experiments to assess what information images provide and what textual cues can guide image interpretation.  We use SVM classifiers for performance, and Multinomial Naive Bayes classifiers to explain classifier decision making, both with bag-of-words features.
\begin{table}[]
\centering
\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|>{\centering\arraybackslash}m{80mm}|}{Q1. Information in text}  \\ \hline
1 & do it clearly          & on which          \\ \hline
2 & let cool for           & favorite toppings \\ \hline
3 & recipe with directions & after an          \\ \hline
4 & how slowly the         & lightly season    \\ \hline
5 & 7 minutes on           & the 2             \\ \hline
\end{tabular}

\vspace{10pt}

\begin{tabular}{|l|l|l|}
\hline
\multicolumn{3}{|>{\centering\arraybackslash}m{80mm}|}{Q1. Information in images  }  \\ \hline
1 & added a beautiful          & cover with    \\ \hline
2 & put as much           & scrapping the        \\ \hline
3 & skin off of            & finally fold        \\ \hline
4 & cut side toward        & after an            \\ \hline
5 & blend and blend        & add a               \\ \hline
\end{tabular}
\vspace{5pt}
\caption{Top five bigram and trigram features of NBSVM for the first question. The highlighted text that is most relevant to the image describes depicted actions, while the complement descriptions describe quantities or modifications of the actions that are described in the highlighted segments.}

\label{table:bigram-trigram}
\end{table}

Table~\ref{svm} reports the F1 measure for instance classification with SVMs (with 5-fold cross validation).  In many cases, machine learning is able to find cues that reliably help guess inferential patterns.
Table~\ref{features} looks at two effective Naive Bayes classifiers, for Q4 (text has quantities) and Q8 (image depicts action in progress).  It shows the features most correlated with the classification decision and their log probability estimates.  For Q4, not surprisingly, numbers and units are positive instances.

More interestingly, verbs of movement and combination are negative instances, perhaps because such steps normally involve material that has already been measured.  For Q8, a range of physical action verbs are associated with actions in progress; negative features correlate with steps involved in actions that don't require ongoing attention (e.g., baking).
Table~\ref{table:bigram-trigram} reports top SVM with NB (NBSVM) \cite{wang2012baselines} features for Q1 that asks subjects to highlight the part of the text that is most related to the image. Action verbs are part of highlighted text, whereas adverbs and quantitative information that cannot be easily depicted in images are part of the remaining segments of the text.
Such correlations set a direction for designing or learning strategies to select when to include imagery.

\section{Conclusions}
\label{conclusion}
In this paper, we have presented the first dataset describing discourse relations across text and imagery.  This data affords theoretical insights into the connection between images and instructional text, and can be used to train classifiers to support automated discourse analysis. Another important contribution of this study is that it presents a discourse annotation scheme for cross-modal data, and establishes that annotations for this scheme can be procured from non-expert contributors via crowd-sourcing.

Our paper sets the agenda for a range of future research.   One obvious example is to extend the approach to other genres of communication with other coherence relations, such as the distinctive coherence of images and caption text \cite{alikhani:sivl19}.  Another is to link coherence relations to the structure of multimodal discourse.  For example, our methods have not yet addressed whether image--text relations have the same kinds of subordinating or coordinating roles that comparable relations have in structuring text discourse \cite{asher2003logics}.
Ultimately, of course, we hope to leverage such corpora to build and apply better models of multimodal communication.

\section{Acknowledgement}
The research presented here is supported by NSF Award IIS-1526723 and  through a fellowship from the Rutgers Discovery Informatics Institute. Thanks to Gabriel Greenberg, Hristiyan Kourtev and the anonymous reviewers for helpful comments. We would also like to thank the Mechanical Turk annotators for their contributions.

\bibliographystyle{unsrt}


\section{Introduction}
\label{sect:introduction}
Due to the application in error-correction in randomized network coding \cite{KK08}, $q$-analogs of combinatorial designs have gained a lot of interest lately.
Arguably the most important open problem in this field is the question of the existence of a $2$-$(7,3,1)_q$ design, as it has the smallest admissible parameter set of a non-trivial $q$-Steiner system with $t \geq 2$.
It is known as a $q$-analog of the Fano plane and has been tackled in many articles~\cite{EV11,HS11,KP14,Met99,MMY95,Tho87,Tho96}.

In this paper we investigate the automorphism group of a putative binary $q$-analog of the Fano plane.
The following result will be proven in Section~\ref{sect:autfano}.
\begin{theorem}\label{tm:main}
A binary $q$-analog of the Fano plane is either rigid or its automorphism group is cyclic of order $2$, $3$ or $4$.
Representing the automorphism group as a subgroup of $\GL(7,2)$, up to conjugacy it is contained in the following list:
\begin{enumerate}[(a)]
\item The trivial group\text{.}
\item The group of order $2$
\[
	\left\langle\left(
	\begin{smallmatrix}
	0& 1& 0& 0& 0& 0& 0 \\
	1& 0& 0& 0& 0& 0& 0 \\
	0& 0& 0& 1& 0& 0& 0 \\
	0& 0& 1& 0& 0& 0& 0 \\
	0& 0& 0& 0& 0& 1& 0 \\
	0& 0& 0& 0& 1& 0& 0 \\
	0& 0& 0& 0& 0& 0& 1
	\end{smallmatrix}\right)\right\rangle\text{.}
\]
\item One of the following two groups of order $3$:
\[
\left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 1& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)
\right\rangle
\quad
\text{and}
\quad
\left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
\item The cyclic group of order $4$
\[
\left\langle
\left(
\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)
\right\rangle\text{.}
\]
\end{enumerate}
\end{theorem}

The idea of eliminating automorphism groups has been
used in the existence problems for other notorious combinatorial objects. Probably the most prominent example is the projective plane of order $10$. In \cite{W79a,W79b} it was shown that the order of an automorphism is either $3$ or $5$. Those two orders had been eliminated later in \cite{AHT80,JT81}, implying that the automorphism group of a projective plane of order $10$ must be trivial.
Finally, the non-existence has been shown in \cite{LTS89} in an extensive computer search.

There are other examples of existence problems where the same idea has been used, that remain still open.
In coding theory, the question of the existence of a binary doubly-even self-dual code with the parameters $[72,36,16]$ was raised more than 40 years ago in \cite{S73}.
Its automorphisms have been heavily investigated in a series of articles.
After the latest result \cite{YY14}, it is known that its automorphism group is either trivial, cyclic of order $2$, $3$ or $5$, or a Klein four group.
In classical design theory, the smallest $v$ for which the existence of a $3$-design of order $v$ is undecided is $16$; indeed a $3$-$(16, 7, 5)$ design is still unknown.
Recent results show that if such design exists, then its automorphism group is either trivial or a $2$-group \cite{E10, N15}.

The proof of Theorem~\ref{tm:main} is partially based on the following more general result on automorphisms of order $2$ of binary $q$-Steiner triple systems, which will be shown in Section~\ref{sect:triple}.
\begin{theorem}\label{thm:aut2}
Let ${\mathcal D}$ be a binary $S_2[2,3,v]$ $q$-Steiner triple system and $A\in\GL(v,2)$ the matrix representation of an automorphism of ${\mathcal D}$ of order $2$.
For $s\in\{1,\ldots,\lfloor v/2\rfloor\}$ let $A_{v,s}$ denote the $v\times v$ block diagonal matrix built from $s$ blocks $\left(\begin{smallmatrix}0 & 1\\1 & 0\end{smallmatrix}\right)$ followed by a $(v-2s)\times (v-2s)$ unit matrix.
\begin{enumerate}[(a)]
\item In the case $v\equiv 1\bmod {6}$, $A$ is conjugate to a matrix $A_{v,s}$ with $3\mid s$.
\item In the case $v\equiv 3\bmod {6}$, $A$ is conjugate to a matrix $A_{v,s}$ with $s\not\equiv 2\bmod {3}$.
\end{enumerate}
\end{theorem}

\section{Preliminaries}
\subsection{Group actions}
Suppose that a finite group $G$ is acting on a finite set $X$.
For $x \in X$ and $\alpha \in G$, the image of $x$ under $\alpha$ is denoted by $x^{\alpha}$.
The set $x^G = \{x^{\alpha} \mathrel{:} \alpha \in G\}$ is called the \emph{orbit of $x$ under $G$} or the \emph{$G$-orbit of $x$}, in short.
The set of all orbits is a partition of $X$.
We say that $x \in X$ is \emph{fixed} by $G$ if $x^{G} = \{x\}$.
So the fixed elements correspond to the orbits of length $1$.
The set $G_x = \{\alpha \in G \,:\, x^{\alpha} = x \}$ is a subgroup of $G$, called the \emph{stabilizer} of $x$ in $G$.
For all $g\in G$ we have
\begin{equation}
	\label{eq:stab_conj}
	G_{x^{g}} = g^{-1} G_x g\text{.}
\end{equation}
In particular the stabilizers of elements in the same orbit are conjugate, and any conjugate subgroup $H$ of $G_x$ is the stabilizer of some element in the orbit of $x$.
The Orbit-Stabilizer Theorem states that $\#x^G = [G : G_x]$,%
\footnote{The symbol $\#$ denotes the cardinality of a set.}
implying that $\#x^G$ divides $\#G$.
The action of $G$ on $X$ extends in the obvious way to an action on the set of subsets of $X$:
For $S\subseteq X$, we set $S^\alpha = \{x^{\alpha} \,:\, x \in S\}$.
For further reading on the theory of finite group actions, see \cite{Ker99}.

\subsection{Linear and semilinear maps}
Let $V,W$ be vector spaces over a field $F$.
A map $f : V \to W$ is called \emph{semilinear} if it is additive and if there is a field automorphism $\sigma\in\Aut(F)$ such that $f(\lambda\mathbf{v}) = \sigma(\lambda) f(\mathbf{v})$ for all $\lambda\in F^\times$ and all $\mathbf{v}\in V$.
The map $f$ is \emph{linear} if and only if $\sigma = \id_F$.
In fact, the \emph{general linear group} $\GL(V)$ of all linear bijections of $V$ is a normal subgroup of the \emph{general semilinear group} $\GammaL(V)$ of all semilinear bijections of $V$.
Moreover, $\GammaL(V)$ decomposes as a semidirect product $\GL(V) \rtimes \Aut(F)$.
In this representation, the natural action on $V$ takes the form $(\phi,\sigma)(\mathbf{v}) = \phi(\sigma(\mathbf{v}))$, where $\sigma(\mathbf{v})$ is the component-wise application of $\sigma$ to the coordinate vector of $\mathbf{v}$ with respect to some fixed basis.

The center $Z(\GL(V))$ of $\GL(V)$ consists of all diagonal maps $\mathbf{v} \mapsto \lambda\mathbf{v}$ with $\lambda\in F^\times$.
It is a normal subgroup of $\GL(V)$ and of $\GammaL(V)$.
The quotient group $\GL(V)/Z(\GL(V))$ is known as the \emph{projective linear group} $\PGL(V)$, and the quotient group $\GammaL(V)/Z(\GL(V))$ is known as the \emph{projective semilinear group} $\PGammaL(V)$.

\subsection{The subspace lattice}
Let $V$ denote a $v$-dimensional vector space over the finite field ${\mathbb F}_q$ with $q$ elements.
For simplicity, a subspace of $V$ of dimension $r$ will be called an \emph{$r$-subspace}.
The set of all $r$-subspaces of $V$ is called the \emph{Gra{\ss}mannian} and is denoted by $\gauss{V}{r}{q}$.
As in projective geometry, the $1$-subspaces of $V$ are called \emph{points}, the $2$-subspaces \emph{lines} and the $3$-subspaces \emph{planes}.
Our focus lies on the case $q = 2$, where the $1$-subspaces $\langle\mathbf{x}\rangle_{{\mathbb F}_2}\in \gauss{V}{1}{2}$ are in one-to-one correspondence with the nonzero vectors $\mathbf{x} \in V\setminus\{\mathbf{0}\}$.
The number of all $r$-subspaces of $V$ is given by
$$
\#\gauss{V}{r}{q} = \gauss{v}{r}{q} = \frac{(q^v-1)\cdots(q^{v-r+1}-1)}{(q^r-1)\cdots(q-1)}.
$$
The set $\mathcal{L}(V)$ of all subspaces of $V$ forms the subspace lattice of $V$.

By the fundamental theorem of projective geometry, for $v\neq 2$ the automorphism group of $\mathcal{L}(V)$ is given by the natural action of $\PGammaL(V)$ on $\mathcal{L}(V)$.
In the case that $q$ is prime, the group $\PGammaL(V)$ reduces to $\PGL(V)$, and for the case of our interest $q = 2$, it reduces further to $\GL(V)$.
After a choice of a basis of $V$, its elements are represented by the invertible $v\times v$ matrices $A$, and the action on $\mathcal{L}(V)$ is given by the vector-matrix-multiplication $\mathbf{v} \mapsto \mathbf{v} A$.

\subsection{Designs}
\begin{definition}
Let $t,v,k$ be integers with $0 \leq t \leq k\leq v$, $\lambda$ another positive integer and $\mathcal{P}$ a set of size $v$.
A $t$-$(v,k,\lambda)$ \emph{(combinatorial) design} ${\mathcal D}$ is a set of $k$-subsets of $\mathcal{P}$ (called \emph{blocks}) such that every $t$-subset of $\mathcal{P}$ is contained in exactly $\lambda$ blocks of ${\mathcal D}$.
When $\lambda = 1$, ${\mathcal D}$ is called a \emph{Steiner system} and denoted by $S(t,k,v)$.
If additionally $t=2$ and $k=3$, ${\mathcal D}$ is called a \emph{Steiner triple system}.
\end{definition}

For the definition of a $q$-analog of a combinatorial design, the subset lattice on $\mathcal{P}$ is replaced by the subspace lattice of $V$.
In particular, subsets are replaced by subspaces and cardinality is replaced by dimension.
So we get \cite{Cam74,Cam74:a}:

\begin{definition}
Let $t,v,k$ be integers with $0 \leq t \leq k\leq v$, $\lambda$ another positive integer and $V$ an ${\mathbb F}_q$-vector space of dimension $v$.
A $t$-$(v,k,\lambda)_q$ \emph{design} ${\mathcal D}$ over the field ${\mathbb F}_q$ is a set of $k$-subspaces of $V$ (called \emph{blocks}) such that every $t$-subspace of $V$ is contained in exactly $\lambda$ blocks of ${\mathcal D}$.
When $\lambda = 1$, ${\mathcal D}$ is called a \emph{$q$-Steiner system} and denoted by $S_q[t,k,v]$.
If additionally $t=2$ and $k=3$, ${\mathcal D}$ is called a \emph{$q$-Steiner triple system}.
\end{definition}

There are good reasons to consider the subset lattice as a subspace lattice over the unary \enquote{field} ${\mathbb F}_1$ \cite{Cohn-2004}.
Thus, classical combinatorial designs can be seen as the limit case $q=1$ of a design over a finite field.
Indeed, quite a few statements about combinatorial designs have a generalization to designs over finite fields, such that the case $q = 1$ reproduces the original statement \cite{BKKL,KL,KP14,NP14}.

One example of such a statement is the following \cite[Lemma~4.1(1)]{Suzuki-1990}:
If ${\mathcal D}$ is a $t$-$(v, k, \lambda_t)_q$ design, then ${\mathcal D}$ is also an $s$-$(v,k,\lambda_s)_q$ for all $s\in\{0,\ldots,t\}$, where
\[
\lambda_s := \lambda_t \frac{\gauss{v-s}{t-s}{q}}{\gauss{k-s}{t-s}{q}}.
\]
In particular, the number of blocks in ${\mathcal D}$ equals
\[
\#{\mathcal D} = \lambda_0 = \lambda_t \frac{\gauss{v}{t}{q}}{\gauss{k}{t}{q}}.
\]
So a necessary condition on the existence of a design with parameters $t$-$(v, k, \lambda)_q$ is that for all $s\in\{0,\ldots,t\}$ the numbers $\lambda \gauss{v-s}{t-s}{q}/\gauss{k-s}{t-s}{q}$ are integers (\emph{integrality conditions}).
In this case, the parameter set $t$-$(v,k,\lambda)_q$ is called \emph{admissible}.
It is further called \emph{realizable} if a $t$-$(v,k,\lambda)_q$ design actually exists.

For designs over finite fields, the action of $\Aut(\mathcal{L}(V)) \cong \PGammaL(V)$ on $\mathcal{L}(V)$ provides a notion of isomorphism.
Two designs in the same ambient space $V$ are called \emph{isomorphic} if they are contained in the same orbit of this action (extended to the power set of $\mathcal{L}(V)$).
The \emph{automorphism group} $\Aut({\mathcal D})$ of a design ${\mathcal D}$ is its stabilizer with respect to this group action.
If $\Aut({\mathcal D})$ is trivial, we will call ${\mathcal D}$ \emph{rigid}.
Furthermore, for $G \leq \PGammaL(V)$, ${\mathcal D}$ will be called $G$-invariant if it is fixed by all $g\in G$ or equivalently, if $G\leq \Aut({\mathcal D})$.
Note that if ${\mathcal D}$ is $G$-invariant, then ${\mathcal D}$ is also $H$-invariant for all subgroups $H \leq G$.

\subsection{Admissibility and realizability}
The question of the realizability of an admissible parameter set is very hard to answer in general.
In the special case $t = 1$, $q$-Steiner systems are called \emph{spreads}.
It is known that the spread parameters $S_q[1,k,v]$ are realizable if and only if they are admissible if and only if $k$ divides $v$.
However for $t\geq 2$, the problem tantalized many researchers \cite{Beu78, BEO+, EV11,Met99,Tho87,Tho96}.
Only recently, the first example of such a $q$-Steiner system was constructed \cite{BEO+}.
It is a $q$-Steiner triple system with the parameters $S_2[2,3,13]$.

As a direct consequence of the integrality conditions, the parameter set of a Steiner triple system $S(2,3,v)$ or a $q$-Steiner triple system $S_q[2,3,v]$ is admissible if and only if $v\equiv 1,3\bmod 6$ and $v\geq 7$.
In the ordinary case $q=1$, the existence question is completely answered by the result that a Steiner triple system is realizable if and only if it is admissible \cite{Kirkman}.
However in the $q$-analog case, our current knowledge is quite sparse.
The only decided case is given by the above mentioned existence of an $S_2[2,3,13]$.

The smallest admissible case of a $q$-Steiner triple system is $S_q[2,3,7]$, whose existence is open for any prime power value of $q$.
It is known as a \emph{$q$-analog of the Fano plane}, since the unique Steiner triple system $S(2,3,7)$ is the Fano plane.
It is worth noting that there are cases of Steiner systems without a $q$-analog, as the famous large Witt design with parameters $S(5,8,24)$ does not have a $q$-analog for any prime power $q$ \cite{KL}.

\subsection{The method of Kramer and Mesner}
The method of Kramer and Mesner \cite{KM76} is a powerful tool for the computational construction of combinatorial designs.
It has been successfully adopted and used for the construction of designs over a finite field \cite{MMY95,BKL05}.
For example, the hitherto only known $q$-analog of a Steiner triple system in \cite{BEO+} has been constructed by this method.
Here we give a short outline, for more details we refer the reader to \cite{BKL05}.
As another computational construction method we mention the use of tactical decompositions~\cite{JT85}, which has been adopted to the $q$-analog case in~\cite{NP14}.

The \emph{Kramer-Mesner matrix} $M_{t,k}^G$ is defined to be the matrix whose rows and columns are indexed by the $G$-orbits on the set $\gauss{V}{t}{q}$ of $t$-subspaces and on the set $\gauss{V}{k}{q}$ of $k$-subspaces of $V$, respectively.
The entry of $M_{t,k}^G$ with row index $T^G$ and column index $K^G$ is defined as $\#\{K'\in K^G \mid T\subseteq K'\}$.
Now there exists a $G$-invariant $t$-$(v,k,\lambda)_q$ design if and only if there is a zero-one solution vector $\mathbf{x}$ of the linear equation system
\begin{equation}\label{eq:km}
M_{t,k}^{G}\mathbf{x}=\lambda \mathbf{1},
\end{equation}
where $\textbf{1}$ denotes the all-one column vector.
More precisely, if $\mathbf{x}$ is a zero-one solution vector of the system~\eqref{eq:km}, a $t$-$(v,k,\lambda)_q$ design is given by the union of all orbits $K^G$ where the corresponding entry in $\mathbf{x}$ equals one.
If $\mathbf{x}$ runs over all zero-one solutions, we get all $G$-invariant $t$-$(v,k,\lambda)_q$ designs in this way.

\subsection{Normal forms for square matrices}
Let $F$ be a field and $V$ a vector space over $F$ of finite dimension $v$.
Two elements of $\GL(V)$ are conjugate if and only if their transformation matrices are conjugate in the matrix group $\GL(v,F)$ if and only if the transformation matrices are similar in the sense of linear algebra.
If $F$ is algebraically closed, representatives of the matrices in $F^{v\times v}$ up to similarity are given by the Jordan normal forms.
In the following, we discuss two common normal forms for the more general case that $F$ is not algebraically closed, the Frobenius normal form and the generalized Jordan normal form.

For a monic polynomial
\[
    f = a_0 + a_1 X + \ldots + a_{n-1} X^{n-1} + X^n\in F[X]\text{,}
\]
the \emph{companion matrix} $A_f$ of $f$ is the $n\times n$ matrix over $F$ defined as
\[
	A_f
	= \begin{pmatrix}
	    0 & 1 & 0 & \cdots & 0 \\
	    \vdots & \ddots & \ddots & \ddots & \vdots \\
	    \vdots & & \ddots & \ddots & 0 \\
	    0 & \cdots & \cdots & 0 & 1 \\
	    -a_0 & -a_1 & -a_2 & \cdots & -a_{n-1}
	\end{pmatrix}\text{.}
\]

It is known that for any square matrix $A$ over $F$ there are unique monic \emph{invariant factors} $f_1 \mid \ldots \mid f_s \in F[X]$ such that $A$ is conjugate to a block diagonal matrix consisting of the blocks $A_{f_1},\ldots,A_{f_s}$.
This matrix is called the \emph{Frobenius normal form} or the \emph{rational normal form} of $A$.
The last invariant factor $f_s$ equals the minimal polynomial of $A$ and the product of all the invariant factors equals the characteristic polynomial of $A$.
The Frobenius normal form corresponds to a decomposition of $V$ into a minimum number of $A$-cyclic subspaces. (A subspace $U$ is called \emph{$A$-cyclic} if there exists a vector $\mathbf{v}\in V$ such that $U = \langle\mathbf{v}, A\mathbf{v}, A^2\mathbf{v},\ldots\rangle_F$.)

By further decomposing the invariant factors into powers of irreducible polynomials, one arrives at an alternative matrix normal form:
For a positive integer $m$ and a monic irreducible polynomial $f$, we define the \emph{Jordan block}
\[
	J_{f,m}
	= \begin{pmatrix}
		A_f & U & \mathbf{0} & \cdots & \mathbf{0} \\
		\mathbf{0}   & A_f & \ddots & \ddots & \vdots \\
		\vdots & \ddots & \ddots & \ddots & \mathbf{0} \\
		\vdots & & \ddots & A_f & U \\
		\mathbf{0} & \cdots & \cdots & \mathbf{0} & A_f
	\end{pmatrix}
\]
with $m$ consecutive blocks $A_f$ on the diagonal.
Here, $\mathbf{0}$ denotes the $n\times n$ zero matrix and $U$ denotes the $n\times n$ matrix whose only nonzero entry is an entry $1$ in the lower left corner.
Each square matrix over $F$ is conjugate to a \emph{(generalized) Jordan normal form}, which is a block diagonal matrix consisting of Jordan blocks $J_{g_1,m_1},\ldots,J_{g_s,m_s}$ with monic irreducible polynomials $g_i$.
Furthermore, the Jordan normal form is unique up to permuting the Jordan blocks.
The Jordan normal form corresponds to a decomposition of $V$ into a maximal number of $A$-invariant subspaces.
For that reason, the matrix blocks of the Jordan normal form are typically smaller than those in the Frobenius normal form.
Depending on the application, this might be an advantage.

\section{Automorphisms of order $2$ of binary $q$-Steiner triple systems}
\label{sect:triple}
In this section we prove Theorem~\ref{thm:aut2}, which is a result on the automorphisms of order $2$ of a binary $q$-Steiner triple system.

\begin{lemma}
	\label{lma:order2}
    In $\GL(v,2)$ there are exactly $\lfloor v/2\rfloor$ conjugacy classes of elements of order $2$.
    Representatives are given by the block-diagonal matrices $A_{v,s}$ with $s\in\{1,\ldots,\lfloor v/2\rfloor\}$, consisting of $s$ consecutive $2\times 2$ blocks $\left(\begin{smallmatrix}0 & 1\\1 & 0\end{smallmatrix}\right)$, followed by a $(v-2s)\times (v-2s)$ unit matrix.
\end{lemma}

\begin{proof}
	Let $A\in\GL(v,2)$.
	The matrix $A$ is of order $2$ if and only if its minimal polynomial is $X^2 - 1 = (X+1)^2$.
	Equivalently, all the invariant factors of $A$ are $X+1$ or $(X+1)^2$, and the invariant factor $(X+1)^2$ must appear at least once.
	So for the number $s$ of the invariant factor $(X+1)^2$, there are the possibilities $s\in\{1,\ldots,\lfloor v/2\rfloor\}$.
	Representatives of the elements of order $2$ are now given by the associated Frobenius normal forms.
	The invariant factor $X+1$ gives $1\times 1$ blocks $(1)$, and the invariant factor $(X+1)^2 = X^2 + 1$ gives $2\times 2$ blocks $\left(\begin{smallmatrix}0 & 1\\1 & 0\end{smallmatrix}\right)$.
	By putting the $1\times 1$ blocks at the end rather than at the beginning, we attain the matrices $A_{v,s}$.
\end{proof}

For a matrix $A$ of order $2$, the unique conjugate $A_{v,s}$ given by Lemma~\ref{lma:order2} will be called the \emph{type} of $A$.
If $A$ is an automorphism of a design $\mathcal{D}$, by equation~\eqref{eq:stab_conj} there is an isomorphic design having the automorphism $A_{v,s}$.
Therefore, for the investigation of the action of $\langle A\rangle$ on $\mathcal{D}$, we may assume $A = A_{v,s}$ without loss of generality.

\begin{lemma}
	\label{lem:fixp}
	Let $A$ be a matrix of order $2$ and type $A_{v,s}$.
	The action of $\langle A\rangle$ partitions the point set $\gauss{{\mathbb F}_2^v}{1}{q}$ into $2^{v-s} - 1$ fixed points and $2^{v-s-1} (2^s - 1)$ orbits of length $2$.
\end{lemma}

\begin{proof}
Me may assume $A = A_{v,s}$.
By the Orbit-Stabilizer Theorem and $\#\langle A_{v,s}\rangle = 2$, all orbits are of length $1$ or $2$.
From the explicit form of the matrices $A_{v,s}$, it is clear that a vector $\mathbf{x} = (x_1,x_2,\ldots,x_v)\in{\mathbb F}_2^v$ is fixed by $A_{v,s}$ if and only if
\[
    x_1 = x_2\text{,}\quad x_3 = x_4\text{,}\ldots\text{,}\quad x_{2s - 1} = x_{2s}\text{.}
\]
Thus, each of these $2^{v-s} - 1$ vectors forms an orbit of length $1$, and the remaining $(2^v - 1) - (2^{v-s} - 1) = 2\times 2^{v-s-1} (2^s - 1)$ vectors are paired into $2^{v-s-1}(2^s - 1)$ orbits of length $2$.
\end{proof}

\begin{example}
	\label{ex:con2:v3}
	A model of the classical Fano plane is given by the points and the planes in $\PG(2,2) = \PG({\mathbb F}_2^3)$.
	Its automorphism group is $\GL(3,2)$.
	By Lemma~\ref{lma:order2}, there is a single conjugacy class of automorphism types of order $2$, represented by the matrix
	\[
		A_{3,1} = \begin{pmatrix}
			0 & 1 & 0 \\
			1 & 0 & 0 \\
			0 & 0 & 1
		\end{pmatrix}\text{.}
	\]
	The action of $\langle A_{3,1}\rangle$ partitions the point set $\gauss{{\mathbb F}_2^3}{1}{2}$ into the $3$ fixed points
	\[
		\langle(0, 0, 1)\rangle_{{\mathbb F}_2}\text{,}\quad
		\langle(1, 1, 0)\rangle_{{\mathbb F}_2}\text{,}\quad
		\langle(1, 1, 1)\rangle_{{\mathbb F}_2}\text{,}
	\]
	and the two orbits of length $2$
	\[
		\{
		\langle(1, 0, 0)\rangle_{{\mathbb F}_2},\;
		\langle(0, 1, 0)\rangle_{{\mathbb F}_2}
		\}
		\quad\text{and}\quad
		\{
		\langle (1, 0, 1)\rangle_{{\mathbb F}_2},\;
		\langle (0, 1, 1)\rangle_{{\mathbb F}_2}
		\}
		\text{.}
	\]
\end{example}

\begin{example}
	\label{ex:con2:v7}
	By Lemma~\ref{lma:order2}, the conjugacy classes of elements of order $2$ in $\GL(7,2)$ are represented by
\begin{align*}
A_{7,1}
& = \left(\begin{smallmatrix}
    0& 1& 0& 0& 0& 0& 0 \\
    1& 0& 0& 0& 0& 0& 0 \\
    0& 0& 1& 0& 0& 0& 0 \\
    0& 0& 0& 1& 0& 0& 0 \\
    0& 0& 0& 0& 1& 0& 0 \\
    0& 0& 0& 0& 0& 1& 0 \\
    0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\text{,}
& A_{7,2}
& = \left(\begin{smallmatrix}
    0& 1& 0& 0& 0& 0& 0 \\
    1& 0& 0& 0& 0& 0& 0 \\
    0& 0& 0& 1& 0& 0& 0 \\
    0& 0& 1& 0& 0& 0& 0 \\
    0& 0& 0& 0& 1& 0& 0 \\
    0& 0& 0& 0& 0& 1& 0 \\
    0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\text{,}
& A_{7,3}
& = \left(\begin{smallmatrix}
    0& 1& 0& 0& 0& 0& 0 \\
    1& 0& 0& 0& 0& 0& 0 \\
    0& 0& 0& 1& 0& 0& 0 \\
    0& 0& 1& 0& 0& 0& 0 \\
    0& 0& 0& 0& 0& 1& 0 \\
    0& 0& 0& 0& 1& 0& 0 \\
    0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}\right)\text{.}
\end{align*}
The number of fixed points is $63$, $31$ and $15$, and the number of orbits of length~$2$ is $32$, $48$ and $56$, respectively.
\end{example}

Now we investigate the automorphisms of order $2$ of an $S_2[2,3,v]$ $q$-Steiner triple system ${\mathcal D}$.
For the remainder of the article, we will assume that $V = {\mathbb F}_2^v$.
This allows us to identify $\GL(V)$ with the matrix group $\GL(v,2)$.

\begin{lemma}\label{lem:fix_2}
Let ${\mathcal D}$ be an $S_2[2,3,v]$ $q$-Steiner system with an automorphism $A$ of order $2$ and type $A_{v,s}$.
Then each block of ${\mathcal D}$ fixed under the action of $\langle A\rangle$ contains either $3$ or $7$ fixed points.
The number of fixed blocks of the first type is
\begin{equation}\label{eq:f3}
F_3 = 2^{v-s-2}(2^s - 1)
\end{equation}
and the number of fixed blocks of the second type is
\begin{equation}\label{eq:f7}
F_7 = \frac{2^{2v-2s-1}+1-3\cdot 2^{v-s-2}(2^s+1)}{21}\text{.}
\end{equation}
\end{lemma}

\begin{proof}
Without restriction $A = A_{v,s}$.
Let ${\mathcal F}$ denote the set of blocks of ${\mathcal D}$ fixed by $G = \langle A_{v,s}\rangle$.
The restriction of $A_{v,s}$ to any fixed block $K\in{\mathcal F}$ is an automorphism of $\mathcal{L}(K)\cong \mathcal{L}({\mathbb F}_2^3)$ whose order divides $2$.
By Example~\ref{ex:con2:v3}, the restriction is either of the unique conjugacy type of order $2$ or the identity.
So the number of fixed points in $K$ is either $3$ or $7$.
Let ${\mathcal F}_3 \subseteq {\mathcal F}$ denote the set of all fixed blocks with $3$ fixed points and ${\mathcal F}_7$ those having $7$ fixed points.

We double count the set $X$ of all pairs $(\{P_1,P_2\},B)$ where $\{P_1,P_2\}$ is a point-orbit of length $2$ and $B$ is a fixed block of ${\mathcal D}$ passing through $P_1$ and $P_2$.
By Lemma~\ref{lem:fixp}, the number of choices for $\{P_1,P_2\}$ is $2^{v-s-1}(2^s - 1)$.
By the design property, there is exactly $\lambda = 1$ block $B$ of ${\mathcal D}$ passing through the line $L = \langle P_1,P_2\rangle_{{\mathbb F}_2}$.
Since $\{P_1,P_2\}$ is fixed under the action of $A_{v,s}$, so is $L$.
This implies that every block $B'$ in the orbit of $B$ passes through $L$, too.
By $\lambda = 1$, we get $B' = B$, so $B$ is a fixed block.
Thus $\#X = 2^{v-s-1}(2^s - 1)$.

Now we first count the possibilities for the fixed block $B$.
Since $B$ contains a point-orbit of length $2$, necessarily $B\in{\mathcal F}_3$.
By Example~\ref{ex:con2:v3}, each such $B$ contains exactly two point-orbits of length $2$.
So $\#X = 2F_3$, which verifies equation~\eqref{eq:f3}.

Now we double count the set $Y$ of all pairs $(\{P_1,P_2\},B)$ where $P_1,P_2$ are two distinct fixed points and $B$ is a fixed block of ${\mathcal D}$ passing through $P_1$ and $P_2$.
By Lemma~\ref{lem:fixp}, the number of choices for $\{P_1,P_2\}$ is $\binom{2^{v-s}-1}{2} = (2^{v-s} - 1)(2^{v-s-1} - 1)$.
As above, there is a single fixed block of ${\mathcal D}$ passing through $P_1$ and $P_2$, which yields $\#Y = (2^{v-s} - 1)(2^{v-s-1} - 1)$.

Vice versa, for $B\in\mathcal{F}_3$ there are $\binom{3}{2} = 3$ choices for $\{P_1,P_2\}$ and for $B\in\mathcal{F}_7$ the number of choices is $\binom{7}{2} = 21$.
So $\#Y = 3F_3 + 21 F_7$.
This shows that $(2^{v-s} - 1)(2^{v-s-1} - 1) = 3F_3 + 21 F_7$.
Replacing $F_3$ by formula~\eqref{eq:f3}, we obtain formula~\eqref{eq:f7}.
\end{proof}

\begin{corollary}
\label{cor:ord2}
Let ${\mathcal D}$ be an $S_2[2,3,v]$ $q$-Steiner system.
\begin{enumerate}[(a)]
	\item In the case $v \equiv 1\bmod 6$, ${\mathcal D}$ does not have an automorphism of type $A_{v,s}$ with $3\nmid s$.
	\item In the case $v \equiv 3\bmod 6$, ${\mathcal D}$ does not have an automorphism of type $A_{v,s}$ with $s\equiv 2\bmod 3$.
\end{enumerate}
\end{corollary}

\begin{proof}
By definition, the number $F_7$ of Lemma~\ref{lem:fix_2} is an integer.
So the numerator of the right hand side of \eqref{eq:f7} must be a multiple of $7$.

We compute its remainder modulo $7$.
The multiplicative order of $2$ modulo $7$ equals $3$, so the remainder only depends on the values of $v$ and $s$ modulo $3$.
We get:
\[
	\begin{array}{c|ccc}
	& s\equiv 0\bmod 3 & s\equiv 1\bmod 3 & s\equiv 2\bmod 3\\
	\hline
	v\equiv 1\bmod 6 & 0 & 1 & 1 \\
	v\equiv 3\bmod 6 & 0 & 0 & -1
	\end{array}
\]
This concludes the proof.
\end{proof}

Now Theorem~\ref{thm:aut2}, which was stated in Section~\ref{sect:introduction}, follows as a combination of Corollary~\ref{cor:ord2} and Lemma~\ref{lma:order2}.

\section{Automorphisms of a binary $q$-analog of the Fano plane}
\label{sect:autfano}
This section is dedicated to the proof of Theorem~\ref{tm:main}.
We prove this assertion by eliminating all other subgroups of $\GL(7,2)$.
This elimination is obtained by a combination of the theoretical results of Section~\ref{sect:triple} and a computer aided search based on the Kramer-Mesner method.

In the following, we shall denote by ${\mathcal D}$ a putative $S_2[2,3,7]$ $q$-Steiner system.
Its automorphism group $\Aut({\mathcal D})$ is a subgroup of the group $\GL(7,2) \cong \PGammaL({\mathbb F}_2^7)$ which has the order
\begin{equation}
\label{eq:ordgl27}
\#\GL(7,2) = 2^{21}\cdot 3^4\cdot 5^1\cdot 7^2\cdot 31^1\cdot 127^1\text{.}
\end{equation}
According to the Sylow theorems, for each prime power $p^r$ with $p^r \mid \#\Aut({\mathcal D})$ there exists a subgroup $G \le \GL(7,2)$ such that ${\mathcal D}$ is $G$-invariant.
Hence for each prime factor $p \in \{2,3,5,7,31,127\}$, we strive to find a (preferably small) exponent $r$ such that there is no subgroup $G \le \GL(7,2)$ of order $p^r$ admitting a $G$-invariant $S_2[2,3,7]$ Steiner system.
Then, we can conclude that $\Aut({\mathcal D})$ is not divisible by $p^r$.

Since the automorphism groups of isomorphic $q$-Steiner systems are conjugate, it is sufficient to restrict the search to representatives of the subgroups of $\GL(7,2)$ up to conjugacy.
Representatives of the conjugacy classes of the elements are given by the Jordan normal forms (with a fixed order of the Jordan blocks).
This provides an efficient way to create the cyclic subgroups up to conjugacy.%
\footnote{We remark that two non-conjugate elements may generate conjugate subgroups.}
For general subgroup generation, the software package Magma is used \cite{BCP97}.

For all the cases where subgroups were excluded computationally using the Kramer-Mesner method, details are given in Table~\ref{tbl:km}.
The column \enquote{group} contains the group in question.
Its isomorphism type as an abstract group is described in column \enquote{type}.
The columns \enquote{$T$-orb} and \enquote{$K$-orb} contain information on the induced partition of the $2$-subsets and $3$-subsets, respectively.
For example, the entry $4^{644} 2^{42} 1^{7}$ for the group $G_{4,1}$ means that $\gauss{{\mathbb F}_2^7}{2}{2}$ is partitioned into $644$ orbits of length $4$, $42$ orbits of length $2$, and $7$ orbits of length $1$.
If a column of the Kramer-Mesner matrix $M_{t,k}^G$ contains an entry $> \lambda = 1$, then the corresponding orbit $K^G$ cannot be contained in ${\mathcal D}$.
So we can remove all those columns and the respective variables from the equation system.
The orbit lengths and the size of the reduced system are given in the table columns \enquote{red. $K$-orb} and \enquote{size}.
The reduced system is fed into a solver based on the dancing links algorithm \cite{Knu00}, which is executed on a single core of an Intel Xeon E3-1275 V2 CPU.
The resulting computation time to show the insolvability of the system is given in the last table column \enquote{runtime}.
An entry \enquote{\emph{open}} means that the solver did not terminate within the time limit of a few days.
There are two possibilities where the insolvability can be seen immediately without the need to run the solver:
If the reduced system remains with a zero row (indicated by \enquote{\emph{zero row!}}) or if it is impossible to get the size $\#{\mathcal D} = 381$ as a sum of the lengths of the $K$-orbits (indicated by \enquote{\emph{orbits!}}).

\subsection{Groups of order $2^r$}
From Theorem~\ref{thm:aut2}, we get:

\begin{lemma}\label{lem:2}
If ${\mathcal D}$ is invariant under an automorphism group $G$ of order $2$, then $G$ is conjugate to the cyclic group
\[
	G_2
	= \langle A_{7,3}\rangle
	= \left\langle\left(
	\begin{smallmatrix}
	0& 1& 0& 0& 0& 0& 0 \\
	1& 0& 0& 0& 0& 0& 0 \\
	0& 0& 0& 1& 0& 0& 0 \\
	0& 0& 1& 0& 0& 0& 0 \\
	0& 0& 0& 0& 0& 1& 0 \\
	0& 0& 0& 0& 1& 0& 0 \\
	0& 0& 0& 0& 0& 0& 1
	\end{smallmatrix}\right)\right\rangle\text{.}
\]
\end{lemma}

\begin{remark}
We attempted to construct a $G_2$-invariant $q$-Steiner triple system $S_2[2,3,7]$ by the Kramer-Mesner method.
The resulting reduced equation system matrix has size $1379 \times 4947$.
Furthermore, from Lemma~\ref{lem:fix_2} we know that ${\mathcal D}$ has exactly $29$ blocks fixed by $G_2$.
However, even with these constraints, the Kramer-Mesner system turned out to be too large to be solved in a reasonable amount of time.
\end{remark}

\begin{lemma}\label{lem:4}
If ${\mathcal D}$ is invariant under a group $G$ of order $4$, then $G$ is a conjugate to the cyclic group
\[
G_{4,1} =
\left\langle
\left(
\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)
\right\rangle
\text{.}
\]
\end{lemma}

\begin{proof}
Using Magma, we got that there are $42$ subgroup classes of $\GL(7,2)$ of order $4$, falling into $7$ cyclic and $35$ non-cyclic ones.
After removing all groups containing a subgroup of order $2$ of a conjugacy type which is excluded by Lemma~\ref{lem:2}, there remains a single cyclic group $G_{4,1}$ and $7$ non-cyclic groups
\begin{align*}
G_{4,2} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} &
G_{4,3} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 1 \\
1 & 1 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} \\
G_{4,4} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} &
G_{4,5} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 \\
1 & 1 & 1 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} \\
G_{4,6} &
= \left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 & 1 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 1 & 1
\end{smallmatrix}\right)
\right\rangle\text{,} &
G_{4,7} & =
\left\langle
A_{7,3},
\left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{ and} \\
G_{4,8} & =
\left\langle
A_{7,3},
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 1 & 1 & 1 \\
0 & 1 & 1 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{.}
\end{align*}
The latter $7$ groups could be excluded computationally.
\end{proof}

\begin{lemma}\label{lem:8}
The design ${\mathcal D}$ is not invariant under a group of order $8$.
\end{lemma}

\begin{proof}
There are $867$ subgroup classes of $\GL(7,2)$ of order $8$.
After removing all groups containing a subgroup of order $2$ or $4$ of a conjugacy type which is excluded by Lemma~\ref{lem:2} or Lemma~\ref{lem:4}, there remains the single cyclic group
\[
G_{8,1} = \left\langle \left(\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&1&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)\right\rangle
\]
and the two quaternion groups
\begin{align*}
G_{8,2} & = \left\langle
\left(\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1&0&1&0&1&1&0 \\
0&1&1&0&0&1&0 \\
0&0&1&0&0&0&1 \\
0&1&0&1&1&0&0 \\
0&0&1&0&1&0&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1 \\
\end{smallmatrix}\right)
\right\rangle\text{ and} \\
G_{8,3} & = \left\langle
\left(\begin{smallmatrix}
1&1&0&0&0&0&0 \\
0&1&1&0&0&0&0 \\
0&0&1&0&0&0&0 \\
0&0&0&1&1&0&0 \\
0&0&0&0&1&1&0 \\
0&0&0&0&0&1&1 \\
0&0&0&0&0&0&1 \\
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1&1&0&0&0&1&0 \\
0&1&0&0&0&0&1 \\
0&0&1&0&0&0&0 \\
1&0&1&1&1&1&0 \\
0&1&1&0&1&0&1 \\
0&0&1&0&0&1&1 \\
0&0&0&0&0&0&1
\end{smallmatrix}\right)
\right\rangle\text{.}
\end{align*}
These $3$ groups are excluded computationally.
\end{proof}

\subsection{Groups of order $3^r$}
\begin{lemma}\label{lem:3}
If ${\mathcal D}$ is invariant under a group $G$ of order $3$, then $G$ is conjugate to one of the cyclic groups
\[
G_{3,1} = \left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 1& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)
\right\rangle
\quad
\text{and}
\quad
G_{3,2} = \left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
\end{lemma}

\begin{proof}
There are $3$ conjugacy classes of subgroups of $\GL(7,2)$ of order $3$.
They are represented by $G_{3,1}$, $G_{3,2}$ and
\[
G_{3,3} = \left\langle
\left(
\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
1& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
The group $G_{3,3}$ could be excluded computationally.
\end{proof}

\begin{lemma}\label{lem:9}
The design ${\mathcal D}$ is not invariant under a group of order $9$.
\end{lemma}

\begin{proof}
There are four conjugacy classes of subgroups of $\GL(7,2)$ of order $9$, a single cyclic and three non-cyclic ones.
After removing all groups containing a subgroup of order $3$ of a conjugacy type which is excluded by Lemma~\ref{lem:3}, there remains the cyclic group
\[
	G_{9,1}
	= \left\langle
	\left(\begin{smallmatrix}
	1 & 0 & 0 & 0 & 0 & 0 & 0 \\
	0 & 0 & 1 & 0 & 0 & 0 & 0 \\
	0 & 0 & 0 & 1 & 0 & 0 & 0 \\
	0 & 0 & 0 & 0 & 1 & 0 & 0 \\
	0 & 0 & 0 & 0 & 0 & 1 & 0 \\
	0 & 0 & 0 & 0 & 0 & 0 & 1 \\
	0 & 1 & 0 & 0 & 1 & 0 & 0
	\end{smallmatrix}\right)
	\right\rangle
\]
and the non-cyclic group
\[
	G_{9,2}
	= \left\langle\left(
	\begin{smallmatrix}
	0 & 1 & 0 & 0 & 0 & 0 & 0 \\
	1 & 1 & 0 & 0 & 0 & 0 & 0 \\
	0 & 0 & 0 & 1 & 0 & 0 & 0 \\
	0 & 0 & 1 & 1 & 0 & 0 & 0 \\
	0 & 0 & 0 & 0 & 0 & 1 & 0 \\
	0 & 0 & 0 & 0 & 1 & 1 & 0 \\
	0 & 0 & 0 & 0 & 0 & 0 & 1
	\end{smallmatrix}\right),
	\left(\begin{smallmatrix}
	1 & 0 & 1 & 1 & 1 & 0 & 0 \\
	0 & 1 & 1 & 0 & 0 & 1 & 0 \\
	0 & 0 & 0 & 0 & 1 & 1 & 0 \\
	0 & 0 & 0 & 0 & 1 & 0 & 0 \\
	0 & 0 & 0 & 1 & 1 & 0 & 0 \\
	0 & 0 & 1 & 1 & 0 & 1 & 0 \\
	0 & 0 & 0 & 0 & 0 & 0 & 1
	\end{smallmatrix}
	\right)\right\rangle\text{.}
\]
These two groups could be excluded computationally.
\end{proof}

\subsection{Groups of other prime power orders}
\begin{lemma}\label{lem:5}
The design ${\mathcal D}$ is not invariant under a group of order $5$.
\end{lemma}

\begin{proof}
As the Sylow $5$-group of $\GL(7,2)$, there is a unique conjugacy class of subgroups of order $5$.
A representative is the cyclic group
\[
G_5 = \left\langle
\left(
\begin{smallmatrix}
0& 0& 0& 1& 0& 0& 0\\
1& 0& 0& 1& 0& 0& 0\\
0& 1& 0& 1& 0& 0& 0\\
0& 0& 1& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
The action of this group partitions the set of $3$-subspaces into a single fixed point and $2362$ orbits of length $5$.
Because of $\#{\mathcal D} = 381 = 76\cdot 5 + 1$, the fixed $3$-subspace must be contained in ${\mathcal D}$.
After fixing the corresponding variable to the value $1$, the solver returned the insolvability of the system.
\end{proof}

\pagebreak

\begin{lemma}\label{lem:7}
The design ${\mathcal D}$ is not invariant under a group of order $7$.
\end{lemma}

\begin{proof}
The general linear group $\GL(7,2)$ has $3$ conjugacy classes of subgroups of order $7$, represented by
\begin{multline*}
G_{7,1} =
\left\langle\left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
1& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 1& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1\\
\end{smallmatrix}\right)\right\rangle
\text{,}\quad
G_{7,2} =
\left\langle\left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
1& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 1& 1& 0& 0\\
0& 0& 0& 0& 0& 0& 1\\
\end{smallmatrix}\right)\right\rangle \\
\text{and}\quad
G_{7,3} =
\left\langle\left(\begin{smallmatrix}
0& 1& 0& 0& 0& 0& 0\\
0& 0& 1& 0& 0& 0& 0\\
1& 0& 1& 0& 0& 0& 0\\
0& 0& 0& 0& 1& 0& 0\\
0& 0& 0& 0& 0& 1& 0\\
0& 0& 0& 1& 0& 1& 0\\
0& 0& 0& 0& 0& 0& 1\\
\end{smallmatrix}\right)\right\rangle
\text{.}
\end{multline*}
For the first two groups, the Kramer-Mesner system is immediately seen to be contradictory.
The third system was excluded computationally.
\end{proof}

\begin{lemma}\label{lem:31}
The design ${\mathcal D}$ is not invariant under a group of order $31$.
\end{lemma}

\begin{proof}
As the Sylow $31$-subgroup, $\GL(7,2)$ has a unique conjugacy class of subgroups of order $31$.
A representative is given by the group
\[
G_{31} = \left\langle\left(
\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 & 0
\end{smallmatrix}
\right)\right\rangle\text{.}
\]
Its action on the set of $3$-subspaces is semiregular, which means that all the orbits are of full length $31$.
Because of $31 \nmid 381 = \#{\mathcal D}$, the design ${\mathcal D}$ cannot arise as a union of a selection of these orbits.
\end{proof}

\begin{lemma}\label{lem:127}
The design ${\mathcal D}$ is not invariant under a group of order $127$.
\end{lemma}

\begin{proof}
As the Sylow $127$-subgroup, $\GL(7,2)$ has a unique conjugacy class of subgroups of order $127$.
Because of $127 = \gauss{7}{1}{2}$, it is the Singer subgroup of $\GL(7,2)$, which has already been excluded in \cite[p.~242]{Tho87}.
\end{proof}

\subsection{Groups of non-prime power order}
The above results on prime-power orders can be combined into the following restriction on the order of $\Aut({\mathcal D})$:
\begin{lemma}
\label{lem:1234612}
	\[
		\#\Aut({\mathcal D})\in\{1,2,3,4,6,12\}
	\]
\end{lemma}

\begin{proof}
	Assume $\#\Aut({\mathcal D}) \neq 1$ and let $q$ be a prime power dividing $\#\Aut({\mathcal D})$.
	By the Sylow theorems, $\Aut({\mathcal D}) \leq \GL(7,2)$ contains a subgroup $G$ of order $q$ and ${\mathcal D}$ is $G$-invariant.
	By Equation~\eqref{eq:ordgl27} and Lemma~\ref{lem:8}, \ref{lem:9}, \ref{lem:5}, \ref{lem:7}, \ref{lem:31}, $\ref{lem:127}$ we get $q\in\{2,3,4\}$.
\end{proof}

\pagebreak

\begin{lemma}\label{lem:6}
The design ${\mathcal D}$ is not invariant under a group of order $6$.
\end{lemma}

\begin{proof}
There are $12$ subgroup classes of $\GL(7,2)$ of order $6$, falling into $6$ cyclic ones and $6$ of isomorphism type $S_3$.
After removing all groups containing a subgroup of order $2$ or $3$ of a conjugacy type which is excluded by Lemma~\ref{lem:2} or Lemma~\ref{lem:3}, there remains the single cyclic group
\[
G_{6,1} = \left\langle \left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1
\end{smallmatrix}\right)\right\rangle
\]
and the two groups of isomorphism type $S_3$
\begin{align*}
G_{6,2} & = \left\langle
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 0
\end{smallmatrix}\right)
\right\rangle\text{ and} \\
G_{6,3} & = \left\langle
\left(\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right),
\left(\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}\right)
\right\rangle\text{.}
\end{align*}
These $3$ groups are excluded computationally.
\end{proof}

\begin{lemma}\label{lem:12}
The design ${\mathcal D}$ is not invariant under a group of order $12$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:6}, we only need to consider subgroups of $\GL(7,2)$ of order $12$ which do not have a subgroup of order $6$.
The only isomorphism type of a group of order $12$ with this property is the alternating group $A_4$.
However, the Sylow $2$-subgroup of $A_4$ is a Klein four group, which is not possible by Lemma~\ref{lem:4}.
\end{proof}

Now Theorem~\ref{tm:main} follows as a combination of Lemmas~\ref{lem:2}, \ref{lem:4}, \ref{lem:1234612}, \ref{lem:6} and \ref{lem:12}.

\begin{table}[t]
\caption{Kramer-Mesner equation systems}
\label{tbl:km}
\noindent
\noindent\resizebox{\linewidth}{!}{
$\begin{array}{llllllll}
G & \text{type} & T\text{-orb} & K\text{-orb} & \text{red. }K\text{-orb} & \text{size} & \text{runtime} \\
\hline
G_{2} & {\mathbb Z}/2{\mathbb Z} & 2^{1288} 1^{91} & 2^{5800} 1^{211} & 2^{4736} 1^{211} & 1379 \times 4947 & \text{\emph{open}} \\
G_{3,1} & {\mathbb Z}/3{\mathbb Z} & 3^{882} 1^{21} & 3^{3930} 1^{21} & 3^{3720} 1^{21} & 903 \times 3741 & \text{\emph{open}} \\
G_{3,2} & {\mathbb Z}/3{\mathbb Z} & 3^{885} 1^{12} & 3^{3925} 1^{36} & 3^{3710} 1^{36} & 897 \times 3746 & \text{\emph{open}} \\
G_{3,3} & {\mathbb Z}/3{\mathbb Z} & 3^{837} 1^{156} & 3^{3875} 1^{186} & 3^{2170} 1^{186} & 993 \times 2356 & <\text{ 1s} \\
G_{4,1} & {\mathbb Z}/4{\mathbb Z} & 4^{644} 2^{42} 1^{7} & 4^{2900} 2^{98} 1^{15} & 4^{2352} 2^{72} 1^{15} & 693 \times 2439 & \text{\emph{open}} \\
G_{4,2} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{616} 2^{84} 1^{35} & 4^{2816} 2^{252} 1^{43} & 4^{2032} 1^{43} & 735 \times 2075 & \text{\emph{zero row!}} \\
G_{4,3} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{616} 2^{84} 1^{35} & 4^{2816} 2^{252} 1^{43} & 4^{1792} 1^{43} & 735 \times 1835 & \text{\emph{zero row!}} \\
G_{4,4} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2824} 2^{228} 1^{59} & 4^{2032} 2^{144} 1^{59} & 735 \times 2235 & <\text{ 1s} \\
G_{4,5} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{616} 2^{84} 1^{35} & 4^{2816} 2^{252} 1^{43} & 4^{1840} 1^{43} & 735 \times 1883 & \text{\emph{zero row!}} \\
G_{4,6} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2816} 2^{252} 1^{43} & 4^{2000} 2^{96} 1^{43} & 735 \times 2139 & \text{\emph{zero row!}} \\
G_{4,7} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2808} 2^{276} 1^{27} & 4^{1600} 2^{144} 1^{27} & 735 \times 1771 & \text{\emph{zero row!}} \\
G_{4,8} & ({\mathbb Z}/2{\mathbb Z})^2 & 4^{608} 2^{108} 1^{19} & 4^{2816} 2^{252} 1^{43} & 4^{1632} 2^{192} 1^{43} & 735 \times 1867 & \text{1398m 57s}\\
G_5 & {\mathbb Z}/5{\mathbb Z} & 5^{532} 1^{7} & 5^{2362} 1^{1} & 5^{2107} 1^{1} & 539 \times 2108 & \text{1977m 20s} \\
G_{6,1} & {\mathbb Z}/6{\mathbb Z} & 6^{428} 3^{29} 2^{4} 1^{4} & 6^{1928} 3^{69} 2^{16} 1^{4} & 6^{1464} 3^{54} 2^{14} 1^{4} & 465 \times 1536 & \text{21m 39s} \\
G_{6,2} & S_3 & 6^{400} 3^{85} 2^{3} 1^{6} & 6^{1862} 3^{201} 2^{13} 1^{10} & 6^{998} 3^{156} 2^{7} 1^{10} & 494 \times 1171 & <\text{ 1s} \\
G_{6,3} & S_3 & 6^{399} 3^{84} 2^{7} 1^{7} & 6^{1863} 3^{204} 2^{7} 1^{7} & 6^{904} 3^{162} 2^{7} 1^{7} & 497 \times 1080 & <\text{ 1s} \\
G_{7,1} & {\mathbb Z}/7{\mathbb Z} & 7^{376} 1^{35} & 7^{1685} 1^{16} & 7^{1200} 1^{16} & 411 \times 1216 & \text{\emph{zero row!}}\\
G_{7,2} & {\mathbb Z}/7{\mathbb Z} & 7^{381} & 7^{1687} 1^{2} & 7^{1620} 1^{2} & 381 \times 1622 & \text{\emph{orbits!}} \\
G_{7,3} & {\mathbb Z}/7{\mathbb Z} & 7^{381} & 7^{1686} 1^{9} & 7^{1668} 1^{9} & 381 \times 1677 & \text{2443m 17s} \\
G_{8,1} & {\mathbb Z}/8{\mathbb Z} & 8^{322} 4^{21} 2^{3} 1^{1} & 8^{1450} 4^{49} 2^{7} 1^{1} & 8^{1144} 4^{34} 2^{6} 1^{1} & 347 \times 1185 & \text{29s} \\
G_{8,2} & Q & 8^{322} 4^{19} 2^{6} 1^{3} & 8^{1450} 4^{42} 2^{21} 1^{1} & 8^{1160} 4^{28} 2^{12} 1^{1} & 350 \times 1201 & \text{8m 14s} \\
G_{8,3} & Q & 8^{322} 4^{18} 2^{9} 1^{1} & 8^{1450} 4^{43} 2^{18} 1^{3} & 8^{1160} 4^{16} 2^{18} 1^{3} & 350 \times 1197 & \text{1m 50s} \\
G_{9,1} & {\mathbb Z}/9{\mathbb Z} & 9^{294} 3^{7} & 9^{1310} 3^{7} & 9^{1177} 3^{7} & 301 \times 1184 & \text{57s} \\
G_{9,2} & ({\mathbb Z}/3{\mathbb Z})^2 & 9^{291} 3^{15} 1^{3} & 9^{1299} 3^{39} 1^{3} & 9^{1077} 3^{27} 1^{3} & 309 \times 1107 & \text{11m 50s} \\
G_{31} & {\mathbb Z}/31{\mathbb Z} & 31^{86} 1^{1} & 31^{381} & 31^{270} & 87 \times 270 & \text{\emph{orbits!}} \\
\end{array}$}
\end{table}

\section{Conclusion}
We have shown that a binary $q$-analog of the Fano plane can only have very few automorphisms.
This result immediately nullifies many natural approaches for the construction, which would inherently imply too much symmetry.

From the point of view of computational complexity though, the vast part of the search space is still untouched, as it consists of the structures without any symmetry.
We believe that further theoretical insight is needed to reduce the complexity to a computationally feasible level.

After all, the question for the existence of a binary $q$-analog of the Fano plane is still wide open.

\section*{Acknowledgements}
The authors are grateful to the organizers of the conference \enquote{Conference on Random Network Codes and Designs over GF$(q)$} held at the Department of Mathematics of Ghent University, Belgium, from September 18--20, 2013, where the first steps towards the present article were initiated.
This conference was organized in the framework of the COST action IC1104, titled \enquote{Random Network Coding and Designs over GF$(q)$}.
Furthermore, we would like to thank the anonymous referees for valuable suggestions improving the readability of the paper.