# A History of Channel Coding in Aeronautical Mobile Telemetry and Deep-Space Telemetry
## 3 Deep-Space Telemetry
### Modern Codes in Future Deep-Space Telemetry  
Fountain codes grew out of the data network world where communication over a binary erasure channel with unknown or quickly changing erasure probabilities was of interest [105]. For a set of \(k\) input symbols, a fountain code produces a potentially limitless set of output symbols where each output symbol is the sum of a subset of the input symbols where subset membership is defined by a distribution. The decoder produces the original set of \(k\) symbols with high probability from an observation comprising \(n\) symbols. Because \(n\) is not fixed, the fountain code is a rateless code. The closer \(n\) is to \(k\), the better the fountain code. The first practical rateless codes were introduced by [PERSON] [106] and called LT codes. Raptor codes are constructed on LT codes by applying a precoder in the form of an outer code [105]. The ability of a rateless code to operate in a variety of channels without a fixed rate has made them of interest in deep-space communications. An interesting example is the study [100] where a joint source coder and channel code, based on Raptor codes over GF(4), were applied to images from the Mars Exploration Rover. The new source/channel coding approach was shown to have advantages over the adaptive rate scheme based on the CCSDS LDPC codes used for the Mars Relay Network.  
A polar code is another form of a rateless code introduced by [PERSON] [107]. Polar codes have recently been adopted as the channel code for the uplink and downlink control information channel in the 5G enhanced mobile broadband (eMBB) service [108]. The absence of \"error floors\", characteristic of LDPC bit error rate performance, together with relatively low complexity decoding algorithms, makes polar codes an attractive option for deep-space communications. A recent study [109] compared polar codes to the CCSDS LDPC codes for deep space optical links. The results were mixed, depending on the  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Rate-1/2** & **Rate-2/3** & **Rate-4/5** \\ \hline \(k=1024\) & \(n=2048\) & \(n=1536\) & \(n=1280\) \\ \(k=4096\) & \(n=8192\) & \(n=6144\) & \(n=5120\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: CCSDS LDPC code parameters.  
Figure 18: A block diagram of the _New Horizons_ telemetry system.  
assumptions adopted for the simulation, but showed that for short block lengths, polar codes outperformed the LDPC codes.  
Channel coding with short block length codes has been of recent interest in machine-type communications, the Internet of Things, and command links, such as those used for deep-space communications [110]. Coding using short block length codes in deep-space communications was examined in [111]. Here, the common approach of including CRC bits to a block of information bits prior to convolutional encoding is reexamined for the case of short block lengths. A list Viterbi algorithm was developed based on the view of a concatenated coding system. The results showed a nearly 3-dB improvement in coding gain relative to the Viterbi algorithm operating only on the convolutional code.