# Low Delay Inter-Packet Coding in Vehicular Networks
## 4 Numerical Results
### Memoryless Channel (BEC)  
As a channel model, we consider the BEC with erasure (packet loss) probability \(\epsilon\). In Figure 3, we compare the probability of packet loss after decoding the Wyner-Ash code by using two decoding methods: the maximum-likelihood SW-decoder (SWML decoder) with delay \(L=4\) and subblock length \(n=4\) and the SWBP-decoder with one redundant parity check (SWBP-RPC decoding) with delays \(L=2\), \(3\), \(4\). We do not show the SWML decoding performance for \(L=2\) and \(3\) since the corresponding plots differ negligibly from the plot for \(L=4\). In addition, we present a lower bound on the decoding performance without restrictions on the decoding delay. This bound is obtained from the spectrum of the Wyner-Ash code presented in Table 1. Only the first term is taken into account, and this gives:  
\[P_{e}\geq 18e^{3}/4. \tag{19}\]  
The plots shown in Figure 3 demonstrate that both SWBP-RPC and SWML decoding provide the packet recovering performance rather close to the lower bound. Therefore, the delay value \(L=4\) is enough for achieving near optimum performance almost in the entire range of packet loss rates \(\epsilon\). In addition, we noticed that the decoding performance is not critical with respect to the decoding delay and \(L=2\) is an acceptable value. Moreover, a very simple SWBP-RPC decoder performs almost as good as the more complicated (yet simple) SWML decoder requiring Gaussian elimination for correcting erasures. In Figure 4, we analyze the coding gain from using SWBP-RPC decoding compared to SWBP decoding. We chose decoding delay \(L=2\) since it follows from the previous plot that increasing delay does not improve the performance significantly. We conclude from this plot that even the SWBP decoder works very well. However, it is worth using the SWBP-RPC decoder because of an additional gain in performance which can be obtained at the cost of negligible increase in decoding complexity. In Figure 5, binary and nonbinary (RS) convolutional coding efficiency is compared. Surprisingly, when the decoding delay is small, \(L=2\), the two codes have near the same efficiency. However, if the allowed decoding delay grows, the RS convolutional codes become more efficient. We also compare the packet loss probability with the lower bound in Equation (16) achievable without restriction on the decoding delay. Again, we can see that the SWML-decoder for the RS convolutional code achieves near optimal performance already for \(L=4\). Notice that, if \(L\) grows, then the decoding complexity grows approximately proportional to \(L^{3}\). It is interesting to compare efficiency of the considered convolutional codes to block codes under the same delay constraints. Delay \(L=2\) and 4 correspond to the length of block codes \(N=12\) and 20, respectively. The block codes of rate \(R_{b}=3/4\) with the best minimum distance for given length and rate were chosen for comparison. Figure 4: Comparison of SWBP, SWBP-RPC and SWML decoding of the binary Wyner–Ash code of rate \(R=3/4\). Figure 3: Comparison of SWML decoding and SWBP decoding with RPC of the binary Wyner–Ash code of rate \(R=3/4\). In Figure 6, the comparison is done for binary codes. The convolutional code with delay \(L=2\) provides almost the same efficiency as the block code with code length \(N=20\). Moreover, typically, if one erasure occurs, then the SWML decoder outputs the result with delay at most 4, whereas the decoder of block code processes the entire codeword of length 20. In Figure 7, we can see that the relation between nonbinary RS-block and RS-convolutional codes differs from that for binary codes. The RS-block code of length 12 loses very little compared to the RS-convolutional code. With increasing delay, the advantage of convolutional codes increases. Notice that the decoding complexity for the convolutional code is less than that for the block codes since its parity-check matrix is more sparse. Figure 5: Comparison of the binary Wyner–Ash code and the RS convolutional code of rate \(R=3/4\).