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Andreas Tsouchlos 2023-03-17 00:33:59 +01:00
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latex/thesis/chapters/decoding_techniques.tex
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@ -199,6 +199,8 @@ of the \acp{LLR} $\gamma_i$ \cite[Sec. 2.5]{feldman_thesis}:%
{f_{Y_i | C_i} \left( y_i \mid C_i = 1 \right) } \right) {f_{Y_i | C_i} \left( y_i \mid C_i = 1 \right) } \right)
.\end{align*} .\end{align*}
% %
\todo{$C_i$ or $c_i$?}%
%
The authors propose the following cost function% The authors propose the following cost function%
\footnote{In this context, \textit{cost function} and \textit{objective function} \footnote{In this context, \textit{cost function} and \textit{objective function}
have the same meaning.} have the same meaning.}
@ -208,6 +210,8 @@ for the \ac{LP} decoding problem:%
g\left( \boldsymbol{c} \right) = \sum_{i=1}^{n} \gamma_i c_i g\left( \boldsymbol{c} \right) = \sum_{i=1}^{n} \gamma_i c_i
.\end{align*} .\end{align*}
% %
\todo{Write as dot product}
%
With this cost function, the exact integer linear program formulation of \ac{ML} With this cost function, the exact integer linear program formulation of \ac{ML}
decoding is the following:% decoding is the following:%
% %
@ -234,6 +238,8 @@ decoding, redefining the constraints in terms of the \text{codeword polytope}
\sum_{\boldsymbol{c} \in \mathcal{C}} \lambda_{\boldsymbol{c}} = 1 \right\} \sum_{\boldsymbol{c} \in \mathcal{C}} \lambda_{\boldsymbol{c}} = 1 \right\}
,\end{align*} % ,\end{align*} %
% %
\todo{$\lambda$ might be confusing here}%
%
which represents the \textit{convex hull} of all possible codewords, which represents the \textit{convex hull} of all possible codewords,
i.e., the convex set of linear combinations of all codewords. i.e., the convex set of linear combinations of all codewords.
This corresponds to simply lifting the integer requirement. This corresponds to simply lifting the integer requirement.