# Memory-Based LT Codes for Efficient 5G Networks and Beyond
## 5 Numerical Results
### DSP and BER  
In this subsection, we investigate the performance of our proposed MBLT codes in the Y-network. We use the proposed scheme over the regular LT codes. We also use the  
Figure 5: DSP versus the overhead (\(\epsilon\)) for LT codes of \(k=256\) at different values of \(c\). (a) \(\delta=0.05\). (b) \(\delta=0.1\). (c) \(\delta=0.2\). (d) \(\delta=0.5\). memory-based concept over one of the well-known distributions in the literature called decreasing ripple size distribution (DRSD) [31]. Thus, our scheme over the DRSD is referred to as memory-based DRSD (MBDRSD). The length of source symbols is set to \(k\) at each source. Both DSP and BER measurements at different \(k\) and overhead \(\epsilon\) over the BEC channel with different erasure probabilities are presented. All the simulation results are run over 100,000 transmissions in each scheme to obtain averages of the implementation measurements. Figure 6 shows the DSP versus the overhead, \(\epsilon=(n-k)/k\), for four different schemes. The figure is obtained at a packet length of \(k=128\), and the optimized parameters of the RSD \(c=0.02\) and \(\delta=0.5\). In order to check the improvement of the proposed algorithm without the effect of the channel characteristics, the erasure probabilities of the three channels in the Y-network are set to zero \(\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}=0\) (i.e., all three channels are clear). As the figure shows, the DSP increases and approaches 100% as the overhead increases (i.e., the number of received symbols increases). However, the faster the scheme approaches 100%, the better the scheme. Moreover, we can observe from the figure that applying the proposed scheme over the traditional LT code with the optimized RSD parameters enhances the performance. In addition, the DRSD scheme outperforms the traditional LT codes with RSD distribution. Nevertheless, applying the concept of memory based on DRSD (i.e., MBDRSD) enhances the performance in terms of the DSP. Although the DSPs of MBRSD and DRSD almost coincide, MBRSD surpasses DRSD in terms of BER, as shown in the following figures. In a similar manner, Figure 7 shows the error probability (i.e., BER) versus the code overhead (\(\epsilon\)) for the four schemes shown in Figure 6 with the same parameters. As has been noted from Figures 6 and 7, the proposed scheme outperforms the conventional LT code and DRSD codes in terms of DSP and BER. However, this improvement comes with costs; the decoding running time is slightly increased. For instance, at the overhead of \(\epsilon=0.4\), the LT decoder requires, on average, 2.07 ms to complete the decoding process with a BER of 0.1227, but the MBLT requires 2.27 ms with a BER of 0.0587. Similarly, DRSD requires 2.29 ms with a BER of \(8.9\times 10^{-3}\), while MBRSD requires 2.49 ms with a BER of \(1.8\times 10^{-3}\). Figure 6: DSP versus the overhead (\(\epsilon\)) for different coding schemes at \(k=128\) and erasure probabilities of \(\varepsilon_{1}=0\), \(\varepsilon_{2}=0\), and \(\varepsilon_{3}=0\). In order to show that the same enhancements are applicable in different environments, the DSP and BER for the same schemes with different channel environments are shown in Figures 8 and 9, respectively. The new channel erasure rates are \(\varepsilon_{1}=0.01\), \(\varepsilon_{2}=0.01\), and \(\varepsilon_{3}=0.1\). These two figures show that using the proposed scheme enhances the performance once it applies to the regular LT codes or DRSD. While the DSP and BER are improved, the decoding time is insignificantly increased. For example, with an overhead of \(\epsilon=0.4\), the regular LT code requires 1.6 ms, and it achieves BER = 0.1553, whereas the MBLT code requires 1.7 ms, and it achieves BER = 0.07836. Likewise, DRSD and MBDRSD require 1.73 ms and 1.87 ms, respectively, and they achieve a BER of 0.0105 and \(3.54\times 10^{-3}\), respectively.