From 4b3d39e605c277beec3069faee573bbfbf8d0405 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Mon, 12 May 2025 23:06:02 +0200 Subject: [PATCH] Fix homotopy definition; Fix simulation results --- Makefile | 4 +- src/2025-03-28/presentation.tex | 464 ++++++++++++++++++-------------- 2 files changed, 260 insertions(+), 208 deletions(-) diff --git a/Makefile b/Makefile index 5512401..f95af6c 100644 --- a/Makefile +++ b/Makefile @@ -4,7 +4,7 @@ all: latexmk src/2024-12-03/presentation.tex mv build/presentation.pdf build/presentation_2024-12-03.pdf - latexmk src/2025-01-07/presentation.tex - mv build/presentation.pdf build/presentation_2025-01-07.pdf + latexmk src/2025-03-28/presentation.tex + mv build/presentation.pdf build/presentation_2025-03-28.pdf clean: rm -rf build diff --git a/src/2025-03-28/presentation.tex b/src/2025-03-28/presentation.tex index d5762f2..f49d6ac 100644 --- a/src/2025-03-28/presentation.tex +++ b/src/2025-03-28/presentation.tex @@ -1,5 +1,5 @@ \documentclass[10pt, aspectratio=169, usenames, dvipsnames]{beamer} - + \usepackage{tikz} \usepackage{tikz-3dplot} \usetikzlibrary{spy, external, intersections} @@ -9,7 +9,6 @@ \pgfplotsset{compat=newest} \usepgfplotslibrary{fillbetween} - \usepackage{listings} \usepackage{subcaption} \usepackage{bbm} @@ -22,27 +21,23 @@ %\definecolor{gruvbox-bg}{HTML}{282828} \definecolor{gruvbox-bg}{HTML}{f2e5bc} - % % % Custom commands % % - \input{lib/latex-common/common.tex} \pgfplotsset{colorscheme/rocket} \newcommand{\res}{src/2025-03-28/res} - % % % CEL Template % % - \newcommand{\templates}{lib/cel-template} \input{\templates/packages.tex} @@ -62,44 +57,38 @@ % \AtBeginSubsubsection[]{} % \AtBeginSubsection[]{} - % % % Set up document % % - \title{HiWi Notes: Minimization of the Code Constraint Polynomial using - Homotopy Continuation Methods} -\subtitle{\small 07.01.2025} +Homotopy Continuation Methods} +\subtitle{\small 28.03.2025} \author{\vspace{1.5mm} Andreas Tsouchlos} \date{ } \institute{Karlsruhe Institute of Technology (KIT), - \\ Communications Engineering Lab (CEL) } +\\ Communications Engineering Lab (CEL) } \tikzstyle{every node}=[font=\small] \captionsetup[sub]{font=small} - % % % Document body % % - \begin{document} - \begin{frame}[plain] - \maketitle + \maketitle \end{frame} \newcommand{\largecitereference}[1] - {\textcolor{kit-green100}{ \large \textbf{{[#1]}} }} - +{\textcolor{kit-green100}{ \large \textbf{{[#1]}} }} % % - The decoding problem as the search for a solution of polynomial system @@ -108,7 +97,6 @@ % - Implementation % - % TODO: Check example homotopy % TODO: Add the name of the homotopy construction (t-1)G + tF and note that it % is not the only possibility @@ -117,27 +105,35 @@ \begin{minipage}[c]{0.65\textwidth} \begin{itemize} - \item Goal: Solve system of equations $F(\bm{x}) = \bm{0}, \hspace{2mm} F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ - \item Problem: Depending on $F$, solving this directly may be difficult + \item Goal: Solve system of equations $F(\bm{x}) = \bm{0}, + \hspace{2mm} F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ + \item Problem: Depending on $F$, solving this directly + may be difficult \item Solution: Define \emph{homotopy function} $H(\bm{x}, t)$ with \begin{gather*} - H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x}, 1) = F(\bm{x}) - ,\end{gather*} - i.e., a deformation between two systems $G(\bm{x})$ and $F(\bm{x})$ + H(\bm{x}, 0) = G(\bm{x}), \hspace{5mm} H(\bm{x}, + 1) = F(\bm{x}) + , + \end{gather*} + i.e., a deformation between two systems $G(\bm{x})$ + and $F(\bm{x})$ (where the zeros of $G$ can be easily obtained); E.g., \begin{gather*} H(\bm{x}, t) = (t-1)G(\bm{x}) + tF(\bm{x}) - .\end{gather*} - Then, compute $(\bm{x}_0, 0)$ such that $G(\bm{x}_0) = \bm{0}$ and trace path to $(\bm{x}_1, 1)$ with $F(\bm{x}_1) = \bm{0}$ + . + \end{gather*} + Then, compute $(\bm{x}_0, 0)$ such that $G(\bm{x}_0) = + \bm{0}$ and trace path to $(\bm{x}_1, 1)$ with + $F(\bm{x}_1) = \bm{0}$ \end{itemize} \vspace{5mm} - \addreferences - {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy + \addreferences + {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy continuation method for solving systems of nonlinear and - polynomial equations}. Communications in Information and - Systems 15.2 (2015): 119-307.} + polynomial equations}. Communications in Information and + Systems 15.2 (2015): 119-307.} \stopreferences \end{minipage}% \hfill @@ -151,70 +147,70 @@ \begin{tikzpicture} \begin{axis}[ - xmin=-1,xmax=1, - ymin=-1,ymax=1, - width=\figlength, - height=\figlength, - ticks=none, - view={0}{90}, - title={$t=0$}, - title style={yshift=-1mm}, - % xlabel={$x_1$}, - % ylabel={$x_2$}, + xmin=-1,xmax=1, + ymin=-1,ymax=1, + width=\figlength, + height=\figlength, + ticks=none, + view={0}{90}, + title={$t=0$}, + title style={yshift=-1mm}, + % xlabel={$x_1$}, + % ylabel={$x_2$}, ] \addplot3[point meta=\thisrow{Hmag}, - point meta min=0, - point meta max=2.5, - quiver={u=\thisrow{H1}, - v=\thisrow{H2}, - scale arrows=.25, - every arrow/.append style={% - line width=.5 - +\pgfplotspointmetatransformed/1000, - -{Latex[length=0pt 5,width=0pt 3]} - }, - }, - quiver/colored = {mapped color}, - -stealth, - ] - table[col sep=comma, discard if not={t}{0.0}] - {\res/H.csv}; + point meta min=0, + point meta max=2.5, + quiver={u=\thisrow{H1}, + v=\thisrow{H2}, + scale arrows=.25, + every arrow/.append style={% + line width=.5 + +\pgfplotspointmetatransformed/1000, + -{Latex[length=0pt 5,width=0pt 3]} + }, + }, + quiver/colored = {mapped color}, + -stealth, + ] + table[col sep=comma, discard if not={t}{0.0}] + {\res/H.csv}; \addplot[mark=*] coordinates {(0,0)} node[above] - {$\bm{x}_0$}; + {$\bm{x}_0$}; \end{axis} \end{tikzpicture} \begin{tikzpicture} \begin{axis}[ - xmin=-1,xmax=1, - ymin=-1,ymax=1, - width=\figlength, - height=\figlength, - ticks=none, - view={0}{90}, - title={$t=0.5$}, - title style={yshift=-1mm}, - % xlabel={$x_1$}, - % ylabel={$x_2$}, + xmin=-1,xmax=1, + ymin=-1,ymax=1, + width=\figlength, + height=\figlength, + ticks=none, + view={0}{90}, + title={$t=0.5$}, + title style={yshift=-1mm}, + % xlabel={$x_1$}, + % ylabel={$x_2$}, ] \addplot3[point meta=\thisrow{Hmag}, - point meta min=0, - point meta max=2.5, - quiver={u=\thisrow{H1}, - v=\thisrow{H2}, - scale arrows=.25, - every arrow/.append style={% - line width=.5 - +\pgfplotspointmetatransformed/1000, - -{Latex[length=0pt 5,width=0pt 3]} - }, - }, - quiver/colored = {mapped color}, - -stealth, - ] - table[col sep=comma, discard if not={t}{0.5}] - {\res/H.csv}; + point meta min=0, + point meta max=2.5, + quiver={u=\thisrow{H1}, + v=\thisrow{H2}, + scale arrows=.25, + every arrow/.append style={% + line width=.5 + +\pgfplotspointmetatransformed/1000, + -{Latex[length=0pt 5,width=0pt 3]} + }, + }, + quiver/colored = {mapped color}, + -stealth, + ] + table[col sep=comma, discard if not={t}{0.5}] + {\res/H.csv}; \draw[line width=1pt] (0,0) -- (0.25,-0.25); \addplot[mark=*] coordinates {(0.25, -0.25)}; @@ -225,48 +221,47 @@ \begin{tikzpicture} \begin{axis}[ - xmin=-1,xmax=1, - ymin=-1,ymax=1, - width=\figlength, - height=\figlength, - ticks=none, - view={0}{90}, - title={$t=1$}, - title style={yshift=-1mm}, - % xlabel={$x_1$}, - % ylabel={$x_2$}, + xmin=-1,xmax=1, + ymin=-1,ymax=1, + width=\figlength, + height=\figlength, + ticks=none, + view={0}{90}, + title={$t=1$}, + title style={yshift=-1mm}, + % xlabel={$x_1$}, + % ylabel={$x_2$}, ] \addplot3[point meta=\thisrow{Hmag}, - point meta min=0, - point meta max=2.5, - quiver={u=\thisrow{H1}, - v=\thisrow{H2}, - scale arrows=.25, - every arrow/.append style={% - line width=.5 - +\pgfplotspointmetatransformed/1000, - -{Latex[length=0pt 5,width=0pt 3]} - }, - }, - quiver/colored = {mapped color}, - -stealth, - ] - table[col sep=comma, discard if not={t}{1.0}] - {\res/H.csv}; + point meta min=0, + point meta max=2.5, + quiver={u=\thisrow{H1}, + v=\thisrow{H2}, + scale arrows=.25, + every arrow/.append style={% + line width=.5 + +\pgfplotspointmetatransformed/1000, + -{Latex[length=0pt 5,width=0pt 3]} + }, + }, + quiver/colored = {mapped color}, + -stealth, + ] + table[col sep=comma, discard if not={t}{1.0}] + {\res/H.csv}; \draw[line width=1pt] (0,0) -- (0.5,-0.5); \addplot[mark=*] coordinates {(0.5,-0.5)} node[below right] - {$\bm{x}_1$}; + {$\bm{x}_1$}; \end{axis} \end{tikzpicture} \caption{Visualization of ``snapshots'' of $H$ (e.g., $F, G$) as - vector fields} + vector fields} \end{figure} \end{minipage} \end{frame} - \begin{frame} \frametitle{Path Tracing} @@ -275,10 +270,13 @@ \begin{itemize} \item Reminder: We are trying to trace the solution curve $H(\bm{x}, t) = \bm{0}$ from $t = 0$ to $t = 1$ - \item We can express the solution curve as a system of differential equations \citereference{CL15}: + \item We can express the solution curve as a system of + differential equations \citereference{CL15}: \begin{align*} DH(\bm{y}(s))\cdot \dot{\bm{y}}(s) &= 0 \\ - \text{det}\left(\begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s)\end{array}\right) &= \sigma_0 \\ + \text{det}\left( + \begin{array}{c} DH(\bm{y}(s)) \\ \dot{\bm{y}}(s) + \end{array}\right) &= \sigma_0 \\ \lVert \dot{\bm{y}}(s) \rVert &= 1 \\ \bm{y}(0) &= (\bm{x}_0, 0) ,% @@ -290,21 +288,23 @@ predictor-corrector scheme, e.g., Euler's predictor and Newton's corrector \citereference{CL15}: \begin{align*} - \hat{\bm{y}} &= \bm{y}_0 + \Delta s \cdot \sigma \cdot {\bm{y}}(\bm{s})\\ - \bm{y} &= \mathcal{N}^k(\hat{\bm{y}}), \hspace{5mm} \mathcal{N}(\hat{\bm{y}}) := \hat{\bm{y}} - (DH(\hat{\bm{y}}))^{+} H(\hat{\bm{y}}) + \hat{\bm{y}} &= \bm{y}_0 + \Delta s \cdot \sigma \cdot + {\bm{y}}(\bm{s})\\ + \bm{y} &= \mathcal{N}^k(\hat{\bm{y}}), \hspace{5mm} + \mathcal{N}(\hat{\bm{y}}) := \hat{\bm{y}} - + (DH(\hat{\bm{y}}))^{+} H(\hat{\bm{y}}) .% \end{align*} \end{itemize} - \addreferences - {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy + \addreferences + {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy continuation method for solving systems of nonlinear and - polynomial equations}. Communications in Information and - Systems 15.2 (2015): 119-307.} + polynomial equations}. Communications in Information and + Systems 15.2 (2015): 119-307.} \stopreferences \end{frame} - \begin{frame} \frametitle{Channel Decoding and Polynomial Equations} @@ -314,36 +314,49 @@ \item To describe the decoding problem we can use the code constraint polynomial \citereference{WT22} \begin{align*} - %h(\bm{x}) = \underbrace{\sum_{i=1}^{n}\left(1-x_i^2\right)^2}_{\text{Bipolar constraint}} + \underbrace{\sum_{j=1}^{m}\left(1 - \left(\prod_{i\in A(j)}x_i\right)\right)^2}_{\text{Parity constraint}} - h(\bm{x}) = \sum_{i=1}^{n}\left(1-x_i^2\right)^2 + \sum_{j=1}^{m}\left(1 - \left(\prod_{i\in A(j)}x_i\right)\right)^2 + %h(\bm{x}) = + % \underbrace{\sum_{i=1}^{n}\left(1-x_i^2\right)^2}_{\text{Bipolar + % constraint}} + \underbrace{\sum_{j=1}^{m}\left(1 - + % \left(\prod_{i\in + % A(j)}x_i\right)\right)^2}_{\text{Parity constraint}} + h(\bm{x}) = \sum_{i=1}^{n}\left(1-x_i^2\right)^2 + + \sum_{j=1}^{m}\left(1 - \left(\prod_{i\in + A(j)}x_i\right)\right)^2 .% \end{align*} - where $A(j) = \left\{i \in [1:n]: \bm{H}_{j,i} = 1\right\}, \hspace{3mm} j\in [1:m]$ + where $A(j) = \left\{i \in [1:n]: \bm{H}_{j,i} = 1\right\}, + \hspace{3mm} j\in [1:m]$ represents the set of variables involved in parity check $j$. \item In a similar vein, we can define a polynomial system whose zeros correspond to codewords as \begin{align*} - F(\bm{x}) = \left[\begin{array}{c}1 - x_1^2 \\ \vdots\\ 1 - x_n^2 \\ 1 - \prod_{i \in A(1)}x_i \\ \vdots\\ 1 - \prod_{i \in A(m)}x_i\end{array}\right] \overset{!}{=} \bm{0} + F(\bm{x}) = \left[ + \begin{array}{c}1 - x_1^2 \\ \vdots\\ 1 - x_n^2 \\ 1 - + \prod_{i \in A(1)}x_i \\ \vdots\\ 1 - + \prod_{i \in A(m)}x_i + \end{array}\right] \overset{!}{=} \bm{0} .% \end{align*} \end{itemize} \addreferences - {WT22}{Tadashi Wadayama; Satoshi Takabe: Proximal Decoding for LDPC Codes. IEICE Transactions on Fundamentals of Electronics, Communi- - cations and Computer Sciences advpub (2022), 2022TAP0002.} + {WT22}{Tadashi Wadayama; Satoshi Takabe: Proximal Decoding for LDPC + Codes. IEICE Transactions on Fundamentals of Electronics, Communi- + cations and Computer Sciences advpub (2022), 2022TAP0002.} \stopreferences \end{frame} - \begin{frame}[fragile] \frametitle{Defining Homotopies for Channel Codes} \begin{itemize} \item Problem: Homotopy continuation algorithms / existing frameworks - only really support square systems, i.e., \# equations = \# variables. The + only really support square systems, i.e., \# equations = + \# variables. The system $F(\bm{x}) = \bm{0}$ we previously considered is overdefined \item \textit{Gröbner bases} allow us to ``[...] transform F into - another set G of polynomials [...] such that F and G are equivalent'' + another set G of polynomials [...] such that F and G are + equivalent'' \citereference{B01}, i.e., they have the same zeros \item Limited tests indicate that, for the systems we are interested in, finding a Gröbner basis yields a square system @@ -352,8 +365,11 @@ \begin{minipage}{0.45\textwidth} \begin{align*} - \overbrace{\bm{H}}^{\text{Parity chek matrix}} &= \left[ \begin{array}{cc} 1 & 1 \end{array}\right] \\ - F(\bm{x}) &= \left[ \begin{array}{c} + \overbrace{\bm{H}}^{\text{Parity check matrix}} &= \left[ + \begin{array}{cc} 1 & 1 + \end{array}\right] \\ + F(\bm{x}) &= \left[ + \begin{array}{c} 1 - x_1^2 \\ 1 - x_2^2\\ 1 - x_1x_2 @@ -367,15 +383,17 @@ \end{minipage}% \begin{minipage}{0.45\textwidth} \begin{align*} - \tilde{F}(\bm{x}) &= \left[ \begin{array}{c} + \tilde{F}(\bm{x}) &= \left[ + \begin{array}{c} x_1 - x_2 \\ x_2^2 - 1 \end{array}\right] \\ - G(\bm{x}) &= \left[\begin{array}{c} - x_1\\ - x_2 + G(\bm{x}) &= \left[ + \begin{array}{c} + x_1 - y_1\\ + x_2 - y_2 \end{array}\right]\\ - H(\bm{x}, t) &= (t-1)G(\bm{x}) + tF(\bm{x}) + H(\bm{x}, t) &= (1-t)G(\bm{x}) + tF(\bm{x}) \end{align*} \end{minipage} \\ @@ -384,11 +402,10 @@ \addreferences {B01}{Buchberger, Bruno. "Gröbner bases: A short introduction for systems theorists." International Conference on Computer Aided Systems Theory. - Berlin, Heidelberg: Springer Berlin Heidelberg, 2001.} + Berlin, Heidelberg: Springer Berlin Heidelberg, 2001.} \stopreferences \end{frame} - \begin{frame}[fragile] \frametitle{Path Tracker Implementation (Pseudo Code)} @@ -409,17 +426,17 @@ baselinestretch=1, % Line spacing breaklines, % Allow line breaking tabsize=4, % Tab width - ]{python} -func perform_prediction_step(y) {...} -func perform_correction_step(y, step_size) {...} + ]{c} +func perform_prediction_step(y, step_size) {...} +func perform_correction_step(y) {...} func perform_step(y0) { for i in range(max_retries): step_size = step_size / 2 - y = perform_prediction_step(y0) + y = perform_prediction_step(y0, step_size) for k in range(max_corrector_iterations): - y = perform_correction_step(y, step_size) + y = perform_correction_step(y) if (corrector converged) break if (corrector converged) break @@ -429,15 +446,14 @@ func perform_step(y0) { \vspace{1mm} - \addreferences - {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy + \addreferences + {CL15}{Chen, Tianran, and Tien-Yien Li.: \emph{Homotopy continuation method for solving systems of nonlinear and - polynomial equations}. Communications in Information and - Systems 15.2 (2015): 119-307.} + polynomial equations}. Communications in Information and + Systems 15.2 (2015): 119-307.} \stopreferences \end{frame} - \begin{frame}[fragile] \frametitle{Decoding Algorithm Implementation (Pseudo Code)} @@ -453,7 +469,7 @@ func perform_step(y0) { baselinestretch=1, % Line spacing breaklines, % Allow line breaking tabsize=4, % Tab width - ]{python} + ]{c} func decode(y) { for i in range(max_iterations): y = perform_step(y) @@ -466,87 +482,123 @@ func decode(y) { \end{minted} \end{frame} - -\begin{frame} +\begin{frame}[fragile] \frametitle{Simulation results} \begin{figure}[H] \centering - \begin{subfigure}{0.5\textwidth} + \begin{subfigure}[c]{0.48\textwidth} \centering + \vskip 0pt \begin{tikzpicture} \begin{axis}[ - domain=-5:5, - width=\textwidth, - height=0.75\textwidth, - ] - \addplot+[mark=none, line width=1pt] - {x^2}; - \addplot+[mark=none, line width=1pt] - {x^3}; - \addplot+[mark=none, line width=1pt] - {x^4}; + width=\textwidth, + height=0.75\textwidth, + ylabel={FER (-\,-\,-), BER (---)}, + ymode=log, + legend, + xlabel = {$Eb/N0$ (dB)}, + legend pos= south west, + xmax=6 + ] + % \addplot+[scol0, densely dashed, mark=none, line + % width=1pt, forget plot] + % table[col sep=comma, x=SNR, y=FER] + % {\res/bch_31_26_hc.csv}; + % \addplot+[scol0, mark=none, line width=1pt] + % table[col sep=comma, x=SNR, y=BER] + % {\res/bch_31_26_hc.csv}; + % \addlegendentry{Homotopy continuation} + % + \addplot+[scol0, densely dashed, mark=none, line width=1pt, + forget plot] + table[col sep=comma, x=SNR, y=FER] + {\res/bch_31_26_hc.csv}; + \addplot+[scol0, mark=none, line width=1pt] + table[col sep=comma, x=SNR, y=BER] + {\res/bch_31_26_hc.csv}; + \addlegendentry{Homotopy continuation} + + \addplot+[scol2, densely dashed, mark=none, line width=1pt, + forget plot] + table[col sep=comma, x=SNR, y=FER, discard if + not={gamma}{0.05}] + {\res/bch_31_26_proximal.csv}; + \addplot+[scol2, mark=none, line width=1pt] + table[col sep=comma, x=SNR, y=BER, discard if + not={gamma}{0.05}] + {\res/bch_31_26_proximal.csv}; + \addlegendentry{Proximal decoding} + + % \addplot+[scol2, densely dashed, mark=none, line + % width=1pt, forget plot] + % table[col sep=comma, x=SNR, y=FER] + % {\res/bch_31_26_ml.csv}; + % \addplot+[scol2, mark=none, line width=1pt] + % table[col sep=comma, x=SNR, y=BER] + % {\res/bch_31_26_ml.csv}; + \addlegendentry{ML} + + % \addplot+[mark=none, line width=1pt] + % table[col sep=comma, x=SNR, y=DFR] + % {\res/bch_31_11.csv}; + % \addlegendentry{DFR} \end{axis} \end{tikzpicture} + + \caption{BCH(31,26) Code} \end{subfigure}% - \begin{subfigure}{0.5\textwidth} + \begin{subfigure}[c]{0.48\textwidth} \centering - \begin{tikzpicture} - \begin{axis}[ - domain=-5:5, - width=\textwidth, - height=0.75\textwidth, - ] - \addplot+[mark=none, line width=1pt] - {x}; - \addplot+[mark=none, line width=1pt] - {2*x}; - \addplot+[mark=none, line width=1pt] - {3*x}; - \end{axis} - \end{tikzpicture} + \vspace*{-20mm} + + \begin{tabular}{rl|ccccc} + Parameter & Value \\ \hline + $n_\text{iter}$ & for homotopy continuation & 20 \\ + $n_\text{iter}$ & for Newton corrector & 5 \\ + $\delta_\text{max}$ & for Newton corrector & 0.01 \\ + $\Delta s$ & for Euler predictor & 0.05 \\ + $n_\text{retries}$ & for Euler predictor & 5 + \end{tabular} + + \bigskip + + \begin{itemize} + \item Newton homotopy: + \begin{align*} + G(\bm{x}) &= F(x) - F(\bm{y}) \\ + H(\bm{x}) &= (1-t)G(\bm{x}) - tF(\bm{x})\\ + &= F(\bm{x}) - (1 - t) F(\bm{y}) + \end{align*} + \end{itemize} \end{subfigure} \end{figure} - - % \begin{figure}[H] - % \centering - % - % \begin{tikzpicture} - % \begin{axis}[ - % width=\textwidth, - % height=0.75\textwidth, - % ] - % \addplot+[mark=none, line width=1pt] - % table[col sep=comma, x=x, y=y] - % {}; - % \end{axis} - % \end{tikzpicture} - % \end{figure} \end{frame} - \begin{frame} \frametitle{Next steps} \begin{itemize} + \item Simulations for other codes \item Thorough investigation into parameter choice - \item Find more mathematical background / guarantees \begin{itemize} - \item How do we have to choose $\sigma_0$? - \item Guarantees for convergence? (i.e., what is the cause for - decoding failures?) - \item When do we actually get square systems using the Gröbner basis? - \end{itemize} - \item Other ideas: \begin{itemize} - \item Generate more candidates by moving further along the - solution curve (if this is possible) and then performing - choosing from this list - \end{itemize} + \item Find more mathematical background / guarantees + \begin{itemize} + \item How do we have to choose $\sigma_0$? + \item Guarantees for convergence? (i.e., what is the cause for + decoding failures?) + \item When do we actually get square systems using + the Gröbner basis? + \end{itemize} + \item Other ideas: + \begin{itemize} + \item Generate more candidates by moving further along the + solution curve (if this is possible) and then performing + choosing from this list + \end{itemize} \end{itemize} \end{frame} - - \end{document}