\chapter{Conclusion and Outlook}%
\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 on their theoretical structure.

For proximal decoding, the effect of each parameter on the behavior of the
decoder was examined, leading to an approach to optimally choose the value
of each parameter.
The convergence properties of the algorithm were investigated in the context
of the relatively high decoding failure rate, to derive an approach to correct
possibly wrong components of the estimate.
Based on this approach, an improvement of 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 by the
\ac{LP} relaxation 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
an efficient implementation of the decoding algorithm.
Based on simulation results, general guidelines for choosing each parameter
were 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 parallels were discovered with regard to the theoretical
structure of the two algorithms, both in the constitution of their respective
objective functions as well 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
a possible cause of its comparatively worse decoding performance.
Furthermore, both algorithms were expressed as message passing algorithms,
illustrating 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.
A different method to improve proximal decoding might be to use
moment-based optimization techniques such as \textit{Adam} \cite{adam}
to try to mitigate the effect of local minima introduced in the objective
function as well as the adversarial structure of the minimization when employing
proximal decoding.

Another area benefiting from future work is the expansion 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}.

