## 5 Investigation and Results
### Bit Error Rate (BER) Performance  
For performance evaluation, BER curves for three conventional decoders and two information bottleneck decoders are shown in Figure 8. All compared systems performed at most \(i=50\) decoding iterations or stopped if the syndrome check was satisfied. The properties of the decoders used are summarized in Table 3.  
The first reference decoder (cf. dash-dotted blue curve in Figure 8) was a belief propagation decoder which receives quasi-continuous LLRs from the channel and performs box-plus operations at the check nodes and summations at the variable node. Clearly, the processing of quasi-continues LLRs in a digital implementation also requires very fine quantization. Standard processor architectures approximate the processing of continuous signal using floating-point or double-precision data types. In a second step, we included a 4-bit channel output quantizer as described in [18] in our simulation. The belief propagation decoder receives only 16 distinct LLRs from the quantizer. However, the internal processing still uses quasi-continuous LLRs. We denote this approach \"belief propagation decoding with quantized channel output\" (cf. dashed green curve in Figure 8). The third considered decoding approach was min-sum decoding (cf. dotted red curve in Figure 8). Just like the belief propagation decoder, the min-sum decoder receives channel LLRs which are not quantized and need to be represented precisely. These LLRs were exchanged during message passing decoding. However, the box-plus operation at the check node was approximated in min-sum decoding and thus much simpler compared to box-plus. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Node Degree**} & \multicolumn{2}{c}{**Direct**} & \multicolumn{2}{c}{**Sequential**} & \multicolumn{2}{c}{**Proposed (Tree-Like)**} \\ \cline{2-7}  & **Entries** & **Look-Ups** & **Entries** & **Look-Ups** & **Entries** & **Look-Ups** \\ \hline \(d_{v}=2\) & 4096 & 2 & 512 & 3 & 512 & 3 \\ \(d_{v}=3\) & \(65,536\) & 3 & 768 & 7 & 512 & 6 \\ \(d_{v}=12\) & \(450.35\cdot 10^{11}\) & 12 & 3072 & 88 & 1280 & 33 \\ \hline \(d_{c}=22\) & \(309.48\cdot 10^{22}\) & 22 & 5376 & 250 & 1536 & 70 \\ \(d_{c}=23\) & \(495.48\cdot 10^{23}\) & 23 & 5632 & 273 & 1536 & 92 \\ \hline Total per Iteration & \(525.931\cdot 10^{23}\) & 62 & 15,360 & 621 & 5376 & 204 \\ \hline Total for \(i=50\) & \(209.2\cdot 10^{27}\) & 3100 & 768,000 & 31,050 & 268,800 & 10,200 \\ \hline Memory for \(i=50\) in kByte & \(104.6\cdot 10^{24}\) & - & 384 & - & 134.4 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overview of the number of look-up table entries and the total memory demand depending on the node structure. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Decoder**} & **Node** & **Messages** & **Messages** & **Message** \\  & **Operations** & **(Internal)** & **(Channel)** & **Alignment** \\ \hline belief propagation & arithmetic & 64 bit & 64 bit & - \\ belief propagation, quantized channel output & arithmetic & 64 bit & 4 bit & - \\ min-sum & approx. arithmetic & 64 bit & 64 bit & - \\ information-bottleneck decoder [10] & look-up Table & 4 bit & 4 bit & no \\ proposed & look-up table & 4 bit & 4 bit & yes \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties of the compared decoding algorithms. The aim of information bottleneck decoders is to combine both domains (i.e., simple node operations and compressed messages) such that only a small number of bits has to be exchanged. We restricted the cardinality of the compression variable \(T\) to 4 bits. The solid magenta curve with \"o\" markers in Figure 8 corresponds to our proposed information bottleneck decoder, which uses message alignment during the construction. When constructed without message alignment, the information bottleneck decoder (cf. solid light-blue curve with \"x\"-markers) diverged already at a BER of \(5\cdot 10^{-3}\).