\chapter{Conclusion}%
\label{chapter:conclusion}

In the context of this thesis, two decoding algorithms were considered:
proximal decoding and \ac{LP} decoding using \ac{ADMM}.
The two algorithms were first analyzed individually, before comparing them
based on simulation results as well as their theoretical structure.

For proximal decoding, the effect of each parameter on the behavior of the
decoder was examined, leading to an approach to choosing the value of each
of the parameters.
The convergence properties of the algorithm were investigated in the context
of the relatively high decoding failure rate, to derive an approach to correct
possible wrong componets of the estimate.
Based on this approach, an improvement over proximal decoding was suggested,
leading to a decoding gain of up to $\SI{1}{dB}$, depending on the code and
the parameters considered.

For \ac{LP} decoding using \ac{ADMM}, the circumstances brought about via the
relaxation while formulating the \ac{LP} decoding problem were first explored.
The decomposable nature arising from the relocation of the constraints into
the objective function itself was recognized as the major driver in enabling
the efficent implementation of the decoding algorithm.
Based on simulation results, general guidelines for choosing each parameter
were again derived.
The decoding performance, in form of the \ac{FER}, of the algorithm was
analyzed, observing that \ac{LP} decoding using \ac{ADMM} nearly reaches that
of \ac{BP}, staying within approximately $\SI{0.5}{dB}$ depending on the code
in question.

Finally, strong parallells were discovered with regard to the theoretical
structure of the two algorithms, both in the constitution of their respective
objective functions as in the iterative approaches used to minimize them.
One difference noted was the approximate nature of the minimization in the
case of proximal decoding, leading to the constraints never being truly
satisfied.
In conjunction with the alternating minimization with respect to the same
variable leading to oscillatory behavior, this was identified as the
root cause of its comparatively worse decoding performance.
Furthermore, both algorithms were expressed as message passing algorithms,
justifying their similar computational performance.

While the modified proximal decoding algorithm presented in section
\ref{sec:prox:Improved Implementation} shows some promising results, further
investigation is required to determine how different choices of parameters
affect the decoding performance.
Additionally, a more mathematically rigorous foundation for determining the
potentially wrong components of the estimate is desirable.
Another area benefiting from future work is the expantion of the \ac{ADMM}
based \ac{LP} decoder into a decoder approximating \ac{ML} performance,
using \textit{adaptive \ac{LP} decoding}.
With this method, the successive addition of redundant parity checks is used
to mitigate the decoder becoming stuck in erroneous solutions introduced due
the relaxation of the constraints of the \ac{LP} decoding problem \cite{alp}.

