\DocumentMetadata{
	lang = en-US,
	pdfstandard = a-4f,
	pdfstandard = ua-2,
	tagging = on,
	tagging-setup = {math/setup={mathml-SE,mathml-AF}}
}

\documentclass{ltx-talk}
\usepackage{unicode-math}
\usepackage{amsthm}

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\ZZ}[1]{\mathbb{Z}/#1\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\RP}{\mathbb{RP}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\p}{\mathfrak{p}}
\newcommand{\q}{\mathfrak{q}}
\newcommand{\m}{\mathfrak{m}}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\Mod}[1]{\ \left(\mathrm{mod}\ #1\right)}
\newcommand{\del}{\partial}
\newcommand{\sm}{\setminus}
\newcommand{\0}{\emptyset}
\newcommand{\cls}[1]{\overline{#1}}
\newcommand{\tsr}[1]{\otimes_{#1}}
\newcommand{\e}{\varepsilon}
\newcommand{\LNorm}[1]{\|#1\|_{L^2[-\pi,\pi]}}
\newcommand{\limup}[1]{\lim_{#1\rightarrow \infty}}
\newcommand{\M}[2]{\mathcal{M}_{#1\times #2}}
\newcommand{\IP}[2]{\langle#1,#2\rangle}
\newcommand*{\fixitem}[1]{\item[]
	\refstepcounter{enumi}\hskip-\labelwidth\hskip-\labelsep
	#1 \labelenumi}
\newcommand{\dbar}{{\mkern3mu\mathchar'26\mkern-12mu d}}

\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\secpder}[2]{\frac{\partial^2 #1}{\partial #2^2}}
\newsavebox{\smallblockbox}

\newenvironment{smallblockarray}
{\begin{lrbox}{\smallblockbox}
		\scriptsize$\begin{blockarray}}
		{\end{blockarray}$\end{lrbox}%
	\raisebox{-1ex}[\dimexpr\height-2ex][\dimexpr\depth-1ex]{\usebox{\smallblockbox}}}

\newcommand{\module}[2]{%
	\underset{\mathclap{\begin{smallmatrix}\\#2\end{smallmatrix}}}{#1}%
}

\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\rad}{rad}
\DeclareMathOperator{\Inn}{Inn}
\DeclareMathOperator{\gap}{gap}
\DeclareMathOperator{\lgap}{lgap}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Jac}{Jac}
\DeclareMathOperator{\Int}{Int}
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\Ass}{Ass}
\DeclareMathOperator{\ch}{char}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\ext}{ext}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator*{\ES}{ES}
\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\eig}{eig}
\DeclareMathOperator{\ev}{ev}
\DeclareMathOperator{\cl}{cl}
\DeclareMathOperator{\Ht}{ht}
\DeclareMathOperator{\inv}{inv}
\DeclareMathOperator{\adj}{adj}
\DeclareMathOperator{\des}{des}
\DeclareMathOperator{\maj}{maj}
\DeclareMathOperator{\Fr}{Fr}
\DeclareMathOperator{\Orb}{Orb}
\DeclareMathOperator{\Stab}{Stab}
\DeclareMathOperator{\Ofr}{OFr}
\DeclareMathOperator{\Map}{Map}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\conv}{conv}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Log}{Log}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\indeg}{indeg}
\DeclareMathOperator{\odeg}{outdeg}
\newcommand{\ds}{\textup{d}_{\rm{s}}}
\renewcommand{\d}{\textup{d}}
\DeclareMathOperator{\Span}{span}
\newcommand{\qbinom}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}}
\newcommand{\dd}{\text{d}_{\textrm{s}}}

\newtheorem*{theorem}{Theorem}
\newtheorem*{corollary}{Corollary}

\title{Weight Distributions of Cyclic Orbit Codes}

\author{Heide Gluesing-Luerssen, \textbf{Hunter Lehmann}}

\institute[UK] {University of Kentucky}

\hypersetup{pdftitle={Weight Distributions of Cyclic Orbit Codes},pdfauthor={Heide Gluesing-Luerssen,Hunter Lehmann}}


\date {January 5, 2022}

\begin{document}
	
	\maketitle % # 1
	
	%%%%%%%%%%%%%%%%%%%%%%
	\section{Introduction and Motivation}
	%%%%%%%%%%%%%%%%%%%%%%
	
	%%
	\begin{frame}
		\frametitle{Subspace Codes}	

		\underline{\textbf{Grassmannian:}}
		\[ \cG_q(k,n)=\{\text{$k$-dim. $\F_q$-subspaces of $\F_{q^n}$}\} \]
		
		\underline{\textbf{Constant dimension subspace code:}}
		\[ \text{A collection of $k$-dimensional $\F_q$-subspaces of $\F_{q^n}$, i.e.} \, \cC \subseteq \cG_q(k,n) \]
		
		
		\underline{\textbf{Subspace distance}:} 
		\begin{align*}
			\dd(\cU,\cV)&=\dim(\cU)+\dim(\cV)-2\dim(\cU\cap \cV) \\
			\dd(\cC)&=\min\limits_{ \cU,\cV \in \cC}\{\d(\cU,\cV) \mid \cU\neq \cV\}
		\end{align*}
		\begin{itemize}
			\item If $\dim(\cU)=\dim(\cV)=k$, $\dd(\cU,\cV)=2k-2\dim(\cU\cap \cV)$ is even.
		\end{itemize}
	\end{frame}
	
	%%
	\begin{frame}
		\frametitle{Cyclic Orbit Codes}
		
		The cyclic (Singer) subgroup $S=\F_{q^n}^* \leq \GL_n(q)$ acts on $\cG_q(k,n)$ by multiplication: for any $\alpha \in \F_{q^n}^*$,
		\[\alpha \cU  :=\{\alpha u \mid u \in \cU \} .\]
		
		A \textbf{cyclic orbit code} is the orbit of a single subspace under this action 
		\[\cC = \Orb_S(\cU)= \{\alpha \cU \mid \alpha \in \F_{q^n}^* \}. \]
		
		The stabilizer of a subspace is always the multiplicative group of a subfield
		\[\Stab_S(\cU)=\F_{q^t}^*, \, \text{for some $t\mid \gcd(n,k)$}. \]
		
		Goal: Find finer invariant for \textbf{optimal cyclic orbit codes}: 
		\[|\Orb_S(\cU)|=\frac{q^n-1}{q-1}\quad \text{and} \quad \dd(\Orb_S(\cU))=2k-2. \]
	\end{frame}
	
	%%
	\begin{frame}
		\frametitle{Projective Spaces}
		
		\underline{\textbf{Projective space:}}
		\[\bP(\F_{q^n})=\F_{q^n}^*/\sim, \] 
		where $\sim$ is the equivalence relation defined by 
		\[a\sim b \Leftrightarrow \frac{a}{b} \in \F_q^*. \]
		
		\pause
		
		\underline{\textbf{Projective subspace:}}  For any $\cU \in \cG_q(k,n)$, define
		\[\bP(\cU)=(\cU\sm\{0\})/\sim\]
		
		
		\textbf{Notation:} We write $\ol{\alpha}$ for the equivalence class of $\alpha$ in $\bP(\F_{q^n})$.
	\end{frame}
	
	%%%%%%%%%%%%%%%%%%%%%%
	\section{Weight Distributions}
	%%%%%%%%%%%%%%%%%%%%%%
	
	%%
	\begin{frame}
		\frametitle{Distributions}
		
		\underline{\textbf{Distance distribution}} of a subspace code $\cC$:
		\[(\delta_0,\delta_1,\ldots,\delta_{d-1},\delta_{d}), \]
		where $\delta_i$ counts the number of pairs $(\cU,\cV)\in \cC \times \cC$ such that $\dd(\cU,\cV)=i.$\\
		
		\vspace{0.5cm}
		
		\underline{\textbf{Weight distribution}} of an orbit code $\cC$ with generator $\cU$:
		\[(d_0,d_2,\ldots,d_{2k-2},d_{2k}), \]
		where $d_i$ counts the number of subspaces $\cV \in \cC$ such that $\dd(\cU,\cV)=i.$\\
		
		\vspace{0.5cm}
		
		\underline{\textbf{Intersection distribution}} of a cyclic orbit code $\cC$ with generator $\cU$:
		\[(\lambda_0,\lambda_1,\ldots,\lambda_\ell),\]
		where $\lambda_i = |\cL_i|=|\{\ol{\alpha} \in \bP(\F_{q^n})\mid \dim(\cU\cap \alpha \cU)=i \}|$.
		
	\end{frame}
	
	%%
	\begin{frame}
		\frametitle{Simplest case: spread codes}
		
		\underline{\textbf{Spread code:}} 
		\[\text{Each} \, \cW \in \cG_q(n,1) \, \text{is contained in exactly one} \,\, \cU \in \cC \]
		Equivalently, for $\cU,\cV \in \cC$:
		
		\[\cU \cap \cV = \{0\} \qquad \text{and} \qquad \bigcup_{\cU\in \cC} \cU =\F_{q^n}.\]
		
		\begin{itemize}
			\item $\dd(\Orb_S(\cU))=2k \Rightarrow \Stab(\cU)=\F_{q^k}^* \, \text{and}\, \Orb_S(\cU) \, \text{is a spread code}$
			\item Intersection distribution is a single entry:
			\[ \lambda_0 = \frac{q^n-q^k}{q-1}.\]
		\end{itemize}
	\end{frame}
	
	%%
	\begin{frame}
		\frametitle{Distance distribution for full-length orbit codes}
		
		\begin{theorem}[Gluesing-Luerssen,L.]
			Let $\cC=\Orb_S(\cU)$ have full-length orbit with $\dd(\cC)=2k-2\ell$ and recall $\cL_i=\{\ol{\alpha} \in \bP(\F_{q^n})\mid \dim(\cU\cap \alpha \cU)=i \}$. Then 
			\[\psi: \{(\overline{u},\overline{v}) \mid \overline{u}\neq \overline{v} \in \bP(\cU))\} \to \bigcup_{i=1}^\ell \cL_i \] by $\psi(\overline{u},\overline{v})=\ol{uv^{-1}}$ is well-defined, surjective, and $\ol{\alpha}\in \cL_i \Leftrightarrow |\psi^{-1}(\ol{\alpha})|=\frac{q^i-1}{q-1}$.
		\end{theorem}
		
		\only<2>{\begin{itemize}
				\item \textbf{Key Idea:} If $\dim(\cU\cap\alpha \cU) \geq 1$, we can write $\ol{\alpha}=\ol{\frac{u}{v}}$ for each equivalence class $\ol{u}$ for $u\in \cU\cap \alpha \cU$.
		\end{itemize}}
		
		\only<3>{\begin{corollary}[Gluesing-Luerssen,L.]
				Let $\cU$ have full-length orbit and $\dd(\Orb_S(\cU))=2k-2\ell$. Then
				\[\sum_{i=1}^\ell \left( \frac{q^i-1}{q-1} \right) \lambda_i = \frac{q^k-1}{q-1}\frac{q^k-q}{q-1}. \]
		\end{corollary}}
		
		
		\only<4>{\begin{corollary}[Gluesing-Luerssen,L.]
				Let $\Orb_S(\cU)$ be an optimal cyclic orbit code. Then $\Orb(\cU)$ has intersection distribution $(\lambda_0,\lambda_1)$, where 
				\[\lambda_1= \frac{q^k-1}{q-1}\frac{q^k-q}{q-1}, \quad \lambda_0=\frac{q^n-1}{q-1}-1-\frac{q^k-1}{q-1}\frac{q^k-q}{q-1}. \]
		\end{corollary}}
		
	\end{frame}
	
	%%
	\begin{frame}
		\frametitle{Full length orbits with smaller minimum distance}
		\begin{theorem}[Gluesing-Luerssen, L.]
			Let $\cU$ have full-length orbit and $\dd(\Orb_S(\cU))=2k-4$. Then the intersection distribution of $\cU$ depends only on $q,n,k$, and a new parameter $r$ describing the orbits of the 2-dimensional intersections $\cU\cap \alpha \cU$.
		\end{theorem}
		
		\pause
		\begin{itemize}
			\item Proof uses group actions, structure of small dimensional intersections, previous Corollary.
			
			\item The theorem does not hold if $\dd(\Orb_S(\cU)) \leq 2k-6$.
			
			\item General behavior of intersection distribution for $\dd(\Orb_S(\cU)) \leq 2k-6$ is open.
		\end{itemize}
	\end{frame}
	
	%%
	\begin{frame}
		\frametitle{Example with $\dd(\cC)=2k-4$}
		$q=2, n=8, k=3$: Fix a generator $\omega$ for $\F_{2^8}^*$ and take $\cU=\langle 1, \omega^{85}, \omega^{17} \rangle$.
		\begin{itemize}
			\only<1>{ \item \textbf{Corollary:} $\frac{q^k-1}{q-1}\frac{q^k-q}{q-1}=\lambda_1 + \frac{q^2-1}{q-1} \lambda_2$}
			\item $42=\lambda_1+3\lambda_2$
			\item \textbf{Q:} What can we learn about $\cV=\cU \cap \alpha \cU$ such that $\dim(\cV)=2$?
			\vspace{0.2cm}
			
			\uncover<2->
			{
				\textbf{Case 1:} $\cV=\F_{2^2}$
				\begin{itemize}
					\item $\ol{\alpha} \in \bP(\F_2^2)\sm\{\ol{1}\}=\{\ol{\omega^{85}},\ol{\omega^{170}} \}$
				\end{itemize}
			}
			\uncover<3->
			{
				\textbf{Case 2:} $\cV$ has full-length orbit, e.g.  $\cV=\langle 1, \omega^{17} \rangle$
				\begin{itemize}
					\item 2 possibilities for $\ol{\alpha}$ e.g. $\ol{\alpha}\in \{ \ol{\omega^{187}}, \ol{\omega^{238}} \}$
					\item Also, $\omega^{-187}\cV = \cU\cap \omega^{-187}\cU$ and $\omega^{-238}\cV= \cU \cap \omega^{-238}\cU$.
					\item $r$ counts the number of these sets (e.g. $\{ \cV, \omega^{-187}\cV, \omega^{-238}\cV \}$) of related $\cV$
				\end{itemize}
			}
			\item<4-> $\lambda_2=q+rq(q+1)=2+2(2)(2+1)=14$
			
			\item<5-> $\lambda_1=42-3\lambda_2=0$
				\item $\lambda_0=\frac{q^n-1}{q-1}-1-\lambda_1-\lambda_2=240$
		\end{itemize}
	\end{frame}


	%%
	\begin{frame}
	\centering{\huge{Thank you.}}
	\vspace{2cm}
	
%	\Large{References}
%	\normalsize{
%	\bibliographystyle{abbrv}
%	\bibliography{../biblio}
%	\nocite{GLL20}}
	\end{frame}
\end{document}