\chapter{Introduction}%
\label{chapter:introduction}

Channel coding using binary linear codes is a way of enhancing the reliability
of data by detecting and correcting any errors that may occur during
its transmission or storage.
One class of binary linear codes, \ac{LDPC} codes, has become especially
popular due to being able to reach arbitrarily small probabilities of error
at code rates up to the capacity of the channel, while retaining a structure
that allows for very efficient decoding.
While the established decoders for \ac{LDPC} codes, such as \ac{BP} and the
\textit{min-sum algorithm}, offer reasonable decoding performance, they are suboptimal
in most cases and exhibit an \textit{error floor} for high \acp{SNR},
making them unsuitable for applications with extreme reliability requiremnts.
Optimization based decoding algorithms are an entirely different way of approaching
the decoding problem, in some cases coming with stronger theoretical guarantees
and promising to alleviate the error floor issue \cite[Sec. I]{original_admm}.

This thesis aims to further the analysis of optimization based decoding
algorithms as well as verify and generalize the considerations present in
the existing literature by considering a variety of different codes.
Specifically, the \textit{proximal decoding} \cite{proximal_paper}
algorithm and \ac{LP} decoding using the \ac{ADMM} \cite{original_admm} are explored.
The two algorithms are analyzed based on their theoretical structure
and based on the results of the simulations conducted in the scope of this work.
Approaches to determine the optimal value of each parameter are derived
and the computational and decoding performance of the algorithms is examined.
An improvement on proximal decoding is suggested, achieving up to $\SI{1}{dB}$
of gain, depending on the parameters chosen and the
code considered.