# A New Hybrid Prime Code for OCDMA Network Multimedia Applications

^{1}

^{2}

^{*}

## Abstract

**:**

^{−9}which is the optimum level.

## 1. Introduction

- (1)
- Maximum number of code sequences for maximum number of users, leading to an increase in the network capacity for data and multimedia applications.
- (2)
- Minimum code length to increase the user bit rate.
- (3)
- Optimal code weight for good network bit error rate (BER) performance.
- (4)
- Minimum cross-correlation values to prevent multiple access interference (MAI).
- (5)
- Reliable code parameter variation while keeping the same cross-correlation value.
- (6)
- Simplicity of code generation.
- (7)
- Minimum cost with respect to the line coding techniques in optical domain.

- (1)
- High signal integrity in the midst of interference at the receiving end.
- (2)
- Broader network capacity.

## 2. Literature Review

## 3. Code Construction

_{1}and P

_{2}; P

_{2}is the greater prime number than P

_{1}. Each prime number can generate a number of code words equal to the same prime number. Each code word contains a number of chips equal to the corresponding prime number. Each code word contains one active chip “HIGH” and the remaining chips are “ZEROs”. The position of the active chip is ${d}_{iorj}$, where ${d}_{iorj}\in \left\{0,1,2,\dots ,{P}_{iorj}-1\right\}$. For the first prime number $P$

_{i}, the code words can be generated as $\left\{{X}_{oi},{X}_{1i},{X}_{2i,}\cdots {X}_{\left({P}_{i}-1\right)i}\right\}$ and for the second prime number P

_{j}, the code words can be generated as $\left\{{X}_{oj},{X}_{1j},{X}_{2j,}\cdots \cdots {X}_{\left({P}_{j}-1\right)j}\right\}$. For example, when P

_{1}= 5 and P

_{2}= 7, the generated code words are listed in the following two groups:

#### 3.1. Code Construction Procedure

- (a)
- Arrange the code sequences in the first tree, row by row in one column, as shown in Table 3, column 1.
- (b)
- (c)
- Rotate the final code sequences in column 2 horizontally from right to left until the first code word in this sequence becomes the last one, as shown in Table 3, column 3.

Column 1 | Column 2 | Column 3 | |||
---|---|---|---|---|---|

Code Index m | First Tree Code Sequences | Merged Code Sequences | Resultant Code Sequences | ||

0 | C_{0} | ${X}_{01}{X}_{11}$ | ${X}_{01}{X}_{02}{X}_{11}{X}_{12}$ | C_{00} | ${X}_{01}$ ${X}_{02}$ ${X}_{11}$ ${X}_{12}$ = 100001000000010000100000 |

C_{01} | ${X}_{12}$ ${X}_{01}$ ${X}_{02}$ ${X}_{11}$ = 010000010000100000001000 | ||||

C_{02} | ${X}_{11}$ ${X}_{12}$ ${X}_{01}$ ${X}_{02}$ = 010000100000100001000000 | ||||

C_{03} | ${X}_{02}$ ${X}_{11}$ ${X}_{12}$ ${X}_{01}$ = 100000001000010000010000 | ||||

1 | C_{1} | ${X}_{01}{X}_{21}$ | ${X}_{01}{X}_{12}{X}_{21}{X}_{22}$ | C_{10} | ${X}_{01}$ ${X}_{12}$ ${X}_{21}$ ${X}_{22}$ = 100000100000001000010000 |

C_{11} | ${X}_{22}$ ${X}_{01}$ ${X}_{12}$ ${X}_{21}$ = 001000010000010000000100 | ||||

C_{12} | ${X}_{21}$ ${X}_{22}$ ${X}_{01}{X}_{12}$ = 001000010000100000100000 | ||||

C_{13} | ${X}_{12}$ ${X}_{21}$ ${X}_{22}$ ${X}_{01}$ = 010000000100001000010000 | ||||

2 | C_{2} | ${X}_{01}{X}_{31}$ | ${X}_{01}{X}_{22}{X}_{31}{X}_{32}$ | C_{20} | ${X}_{01}$ ${X}_{22}$ ${X}_{31}$ ${X}_{32}$ = 100000010000000100001000 |

C_{21} | ${X}_{32}$ ${X}_{01}$ ${X}_{22}$ ${X}_{31}$ = 000100010000001000000010 | ||||

C_{22} | ${X}_{31}$ ${X}_{32}$ ${X}_{01}$ ${X}_{22}$ = 000100001000100000010000 | ||||

C_{23} | ${X}_{22}$ ${X}_{31}$ ${X}_{32}$ ${X}_{01}$ = 001000000010000100010000 | ||||

3 | C_{3} | ${X}_{01}{X}_{41}$ | ${X}_{01}{X}_{32}{X}_{41}{X}_{42}$ | C_{30} | ${X}_{01}$ ${X}_{32}$ ${X}_{41}$ ${X}_{42}$ = 100000001000000010000100 |

C_{31} | ${X}_{42}$ ${X}_{01}$ ${X}_{32}$ ${X}_{41}$ = 000010010000000100000001 | ||||

C_{32} | ${X}_{41}$ ${X}_{42}$ ${X}_{01}$ ${X}_{32}$ = 000010000100100000001000 | ||||

C_{33} | ${X}_{32}$ ${X}_{41}$ ${X}_{42}$ ${X}_{01}$ = 000100000001000010010000 | ||||

4 | C_{4} | ${X}_{11}{X}_{21}$ | ${X}_{11}{X}_{42}{X}_{21}{X}_{52}$ | C_{40} | ${X}_{11}$ ${X}_{42}$ ${X}_{21}$ ${X}_{52}$ = 010000000100001000000010 |

C_{41} | ${X}_{52}$ ${X}_{11}$ ${X}_{42}$ ${X}_{21}$ = 000001001000000010000100 | ||||

C_{42} | ${X}_{21}$ ${X}_{52}$ ${X}_{11}$ ${X}_{42}$ = 001000000010010000000100 | ||||

C_{43} | ${X}_{42}$ ${X}_{21}$ ${X}_{52}$ ${X}_{11}$ = 000010000100000001001000 | ||||

5 | C_{5} | ${X}_{11}{X}_{31}$ | ${X}_{11}{X}_{52}{X}_{31}{X}_{62}$ | C_{50} | ${X}_{11}$ ${X}_{52}$ ${X}_{31}$ ${X}_{62}$ = 010000000010000100000001 |

C_{51} | ${X}_{62}$ ${X}_{11}$ ${X}_{52}$ ${X}_{31}$ = 000000101000000001000010 | ||||

C_{52} | ${X}_{31}$ ${X}_{62}$ ${X}_{11}$ ${X}_{52}$ = 000100000001010000000010 | ||||

C_{53} | ${X}_{52}$ ${X}_{31}$ ${X}_{62}$ ${X}_{11}$ = 000001000010000000101000 | ||||

6 | C_{6} | ${X}_{11}{X}_{41}$ | ${X}_{11}{X}_{02}{X}_{41}{X}_{22}$ | C_{60} | ${X}_{11}$ ${X}_{02}$ ${X}_{41}$ ${X}_{22}$ = 010001000000000010010000 |

C_{61} | ${X}_{22}$ ${X}_{11}$ ${X}_{02}$ ${X}_{41}$ = 001000001000100000000001 | ||||

C_{62} | ${X}_{41}$ ${X}_{22}$ ${X}_{11}$ ${X}_{02}$ = 000010010000010001000000 | ||||

C_{63} | ${X}_{02}$ ${X}_{41}$ ${X}_{22}$ ${X}_{11}$ = 100000000001001000001000 | ||||

7 | C_{7} | ${X}_{21}{X}_{31}$ | ${X}_{21}{X}_{12}{X}_{31}{X}_{32}$ | C_{70} | ${X}_{21}$ ${X}_{12}$ ${X}_{31}$ ${X}_{32}$ = 001000100000000100001000 |

C_{71} | ${X}_{32}$ ${X}_{21}$ ${X}_{12}$ ${X}_{31}$ = 000100000100010000000010 | ||||

C_{72} | ${X}_{31}$ ${X}_{32}$ ${X}_{21}$ ${X}_{12}$ = 000100001000001000100000 | ||||

C_{73} | ${X}_{12}$ ${X}_{31}$ ${X}_{32}$ ${X}_{21}$ = 010000000010000100000100 | ||||

8 | C_{8} | ${X}_{21}{X}_{41}$ | ${X}_{21}{X}_{22}{X}_{41}{X}_{42}$ | C_{80} | ${X}_{21}$ ${X}_{22}$ ${X}_{41}$ ${X}_{42}$ = 001000010000000010000100 |

C_{81} | ${X}_{42}$ ${X}_{21}$ ${X}_{22}$ ${X}_{41}$ = 000010000100001000000001 | ||||

C_{82} | ${X}_{41}$ ${X}_{42}$ ${X}_{21}$ ${X}_{22}$ = 000010000100001000010000 | ||||

C_{83} | ${X}_{22}$ ${X}_{41}$ ${X}_{42}$ ${X}_{21}$ = 001000000001000010000100 | ||||

9 | C_{9} | ${X}_{31}{X}_{41}$ | ${X}_{31}{X}_{32}{X}_{41}{X}_{52}$ | C_{90} | ${X}_{31}$ ${X}_{32}$ ${X}_{41}$ ${X}_{52}$ = 000100001000000010000010 |

C_{91} | ${X}_{52}$ ${X}_{31}$ ${X}_{32}$ ${X}_{41}$ = 000001000010000100000001 | ||||

C_{92} | ${X}_{41}$ ${X}_{52}$ ${X}_{31}$ ${X}_{32}$ = 000010000010000100001000 | ||||

C_{93} | ${X}_{32}$ ${X}_{41}$ ${X}_{52}$ ${X}_{31}$ = 000100000001000001000010 | ||||

10 | C_{10} | ${X}_{11}{X}_{01}$ | ${X}_{11}{X}_{42}{X}_{01}{X}_{62}$ | C_{100} | ${X}_{11}$ ${X}_{42}$ ${X}_{01}$ ${X}_{62}$ = 010000000100100000000001 |

C_{101} | ${X}_{62}$ ${X}_{11}$ ${X}_{42}$ ${X}_{01}$ = 000000101000000010010000 | ||||

C_{102} | ${X}_{01}$ ${X}_{62}$ ${X}_{11}$ ${X}_{42}$ = 100000000001010000000100 | ||||

C_{103} | ${X}_{42}$ ${X}_{01}$ ${X}_{62}$ ${X}_{11}$ = 000010010000000000101000 | ||||

11 | C_{11} | ${X}_{21}{X}_{01}$ | ${X}_{21}{X}_{02}{X}_{01}{X}_{32}$ | C_{110} | ${X}_{21}$ ${X}_{02}$ ${X}_{01}$ ${X}_{32}$ = 001001000000100000001000 |

C_{111} | ${X}_{32}$ ${X}_{21}$ ${X}_{02}$ ${X}_{01}$ = 000100000100100000010000 | ||||

C_{112} | ${X}_{01}$ ${X}_{32}$ ${X}_{21}$ ${X}_{02}$ = 100000001000001001000000 | ||||

C_{113} | ${X}_{02}$ ${X}_{01}$ ${X}_{32}$ ${X}_{21}$ = 100000010000000100000100 | ||||

12 | C_{12} | ${X}_{31}{X}_{01}$ | ${X}_{31}{X}_{12}{X}_{01}{X}_{42}$ | C_{120} | ${X}_{31}$ ${X}_{12}$ ${X}_{01}$ ${X}_{42}$ = 000100100000100000000100 |

C_{121} | ${X}_{42}$ ${X}_{31}$ ${X}_{12}$ ${X}_{01}$ = 000010000010010000010000 | ||||

C_{122} | ${X}_{01}$ ${X}_{42}$ ${X}_{31}$ ${X}_{12}$ = 100000000100000100100000 | ||||

C_{123} | ${X}_{12}$ ${X}_{01}$ ${X}_{42}$ ${X}_{31}$ = 010000010000000010000010 | ||||

13 | C_{13} | ${X}_{41}{X}_{01}$ | ${X}_{41}{X}_{22}{X}_{01}{X}_{52}$ | C_{130} | ${X}_{41}$ ${X}_{22}$ ${X}_{01}$ ${X}_{52}$ = 000010010000100000000010 |

C_{131} | ${X}_{52}$ ${X}_{41}$ ${X}_{22}$ ${X}_{01}$ = 000001000001001000010000 | ||||

C_{132} | ${X}_{01}$ ${X}_{52}$ ${X}_{41}$ ${X}_{22}$ = 100000000010000010010000 | ||||

C_{133} | ${X}_{22}$ ${X}_{01}$ ${X}_{52}$ ${X}_{41}$ = 001000010000000001000001 | ||||

14 | C_{14} | ${X}_{21}{X}_{11}$ | ${X}_{21}{X}_{32}{X}_{11}{X}_{62}$ | C_{140} | ${X}_{21}$ ${X}_{32}$ ${X}_{11}$ ${X}_{62}$ = 001000001000010000000001 |

C_{141} | ${X}_{62}$ ${X}_{21}$ ${X}_{32}$ ${X}_{11}$ = 000000100100000100001000 | ||||

C_{142} | ${X}_{11}$ ${X}_{62}$ ${X}_{21}$ ${X}_{32}$ = 010000000001001000001000 | ||||

C_{143} | ${X}_{32}$ ${X}_{11}$ ${X}_{62}$ ${X}_{21}$ = 000100001000000000100100 | ||||

15 | C_{15} | ${X}_{31}{X}_{11}$ | ${X}_{31}{X}_{02}{X}_{11}{X}_{42}$ | C_{150} | ${X}_{31}$ ${X}_{02}$ ${X}_{11}$ ${X}_{42}$ = 000101000000010000000100 |

C_{151} | ${X}_{42}$ ${X}_{31}$ ${X}_{02}$ ${X}_{11}$ = 000010000010100000001000 | ||||

C_{152} | ${X}_{11}$ ${X}_{42}$ ${X}_{31}$ ${X}_{02}$ = 010000000100000101000000 | ||||

C_{153} | ${X}_{02}$ ${X}_{11}$ ${X}_{42}$ ${X}_{31}$ = 100000001000000010000010 | ||||

16 | C_{16} | ${X}_{41}{X}_{11}$ | ${X}_{41}{X}_{12}{X}_{11}{X}_{52}$ | C_{160} | ${X}_{41}$ ${X}_{12}$ ${X}_{11}$ ${X}_{52}$ = 000010100000010000000010 |

C_{161} | ${X}_{52}$ ${X}_{41}$ ${X}_{12}$ ${X}_{11}$ = 000001000001010000001000 | ||||

C_{162} | ${X}_{11}$ ${X}_{52}$ ${X}_{41}$ ${X}_{12}$ = 010000000010000010100000 | ||||

C_{163} | ${X}_{12}$ ${X}_{11}$ ${X}_{52}$ ${X}_{41}$ = 010000001000000001000001 | ||||

17 | C_{17} | ${X}_{31}{X}_{21}$ | ${X}_{31}{X}_{22}{X}_{21}{X}_{62}$ | C_{170} | ${X}_{31}$ ${X}_{22}$ ${X}_{21}$ ${X}_{62}$ = 000100010000001000000001 |

C_{171} | ${X}_{62}$ ${X}_{31}$ ${X}_{22}$ ${X}_{21}$ = 000000100010001000000100 | ||||

C_{172} | ${X}_{21}$ ${X}_{62}$ ${X}_{31}$ ${X}_{22}$ = 001000000001000100010000 | ||||

C_{173} | ${X}_{22}$ ${X}_{21}$ ${X}_{62}$ ${X}_{31}$ = 001000000100000000100010 | ||||

18 | C_{18} | ${X}_{41}{X}_{21}$ | ${X}_{41}{X}_{02}{X}_{21}{X}_{52}$ | C_{180} | ${X}_{41}$ ${X}_{02}$ ${X}_{21}$ ${X}_{52}$ = 000011000000001000000010 |

C_{181} | ${X}_{52}$ ${X}_{41}$ ${X}_{02}$ ${X}_{21}$ = 000001000001100000000100 | ||||

C_{182} | ${X}_{21}$ ${X}_{52}$ ${X}_{41}{X}_{02}$ = 001000000010000011000000 | ||||

C_{183} | ${X}_{02}$ ${X}_{21}$ ${X}_{52}{X}_{41}$ = 100000000100000001000001 | ||||

19 | C_{19} | ${X}_{41}{X}_{31}$ | ${X}_{41}{X}_{12}{X}_{31}{X}_{62}$ | C_{190} | ${X}_{41}$ ${X}_{12}$ ${X}_{31}$ ${X}_{62}$ = 000010100000000100000001 |

C_{191} | ${X}_{62}$ ${X}_{41}$ ${X}_{12}$ ${X}_{31}$ = 000000100001010000000010 | ||||

C_{192} | ${X}_{31}$ ${X}_{62}$ ${X}_{41}$ ${X}_{12}$ = 000100000001000010100000 | ||||

C_{193} | ${X}_{12}$ ${X}_{31}$ ${X}_{62}$ ${X}_{41}$ = 010000000010000000100001 |

- Limited cross-correlation “0” or “1”;
- Very large number of code sequences can provide a large number of simultaneous users without sacrificing performance;
- Shorter code length for the same higher bit rate transmission.

_{2}= 7.

#### 3.2. Correlation Properties

- (a)
- The peak value of the auto-correlation property is 2n, where n is an integer number equal to the number of code words used to construct the code sequence in each tree.
- (b)
- The value of the cross-correlation property is “0” or “1” between any two different code sequences in the coding of Table 3 and is independent of whether these two codes share the same code index or not.
- (c)
- $m\in \left\{0,1,2,\dots \dots ,\left(Pi+2Pj\right)\right\}$ and $z\in \left\{0,1,\dots ,({P}_{i}-2)\right\}$

## 4. Correlation Results

## 5. The Proposed OCDMA System Model

## 6. BER Performance Analysis

- Code length $L$ equal to $L=2\sum _{i=1}^{n}{P}_{i}$, where ${P}_{i}$ is the prime number and n is the number of the different prime numbers used for the code construction.

_{min}= max(N + P

_{1}− K, 1), and u

_{max}= min(N, P

_{1}).

_{j}is a random variable representing the number of pulses interfering the time slot number j. Additionally, w = (w

_{0}, w

_{1}, ……., w

_{M−}

_{1})

^{U}is the vector that realizes the vector l and U = u.

## 7. Throughput Analysis

## 8. EVM Analysis

## 9. Simulation Results

^{−10}. This means lower MAI as a result of good code correlation characteristics. If the code length and code weight are increased, the system can accommodate a higher number of active users at slightly higher BER due to the incremental increase in the MAI. Despite this increase, the proposed code is still better than the other codes as shown in Figure 22.

## 10. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 8.**Cross-correlation between ${C}_{11}\mathrm{and}{C}_{13}$ sequences in same group, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 9.**Cross-correlation between ${C}_{40}\mathrm{and}{C}_{42}$ sequences in same group, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 10.**Cross-correlation between ${C}_{71}\mathrm{and}{C}_{73}$ sequences in same group, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 11.**Cross-correlation between ${C}_{110}\mathrm{and}{C}_{112}$ sequences in same group, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 12.**Cross-correlation between ${C}_{160}\mathrm{and}{C}_{162}$ sequences in same group, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 13.**Cross-correlation between ${C}_{191}\mathrm{and}{C}_{193}$ sequences in same group, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 14.**Cross-correlation between ${C}_{03}\mathrm{and}{C}_{31}$ sequences from different groups, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 15.**Cross-correlation between ${C}_{21}\mathrm{and}{C}_{43}$ sequences from different groups, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 16.**Cross-correlation between ${C}_{60}\mathrm{and}{C}_{82}$ sequences from different groups, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 17.**Cross-correlation between ${C}_{92}\mathrm{and}{C}_{110}$ sequences from different groups, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 18.**Cross-correlation between ${C}_{121}\mathrm{and}{C}_{150}$ sequences from different groups, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

**Figure 19.**Cross-correlation between ${C}_{173}\mathrm{and}{C}_{193}$ sequences from different groups, ${P}_{1}=5\mathrm{and}{P}_{2}=7$.

Folded Code Sequences | Code Sequences | ||||||
---|---|---|---|---|---|---|---|

${X}_{41}{X}_{01}$ | ${X}_{31}{X}_{01}$ | ${X}_{21}{X}_{01}$ | ${X}_{11}{X}_{01}$ | ${X}_{01}{X}_{11}$ | ${X}_{01}{X}_{21}$ | ${X}_{01}{X}_{31}$ | ${X}_{01}{X}_{41}$ |

${X}_{41}{X}_{11}$ | ${X}_{31}{X}_{11}$ | ${X}_{21}{X}_{11}$ | ${X}_{11}{X}_{21}$ | ${X}_{11}{X}_{31}$ | ${X}_{11}{X}_{41}$ | ||

${X}_{41}{X}_{21}$ | ${X}_{31}{X}_{21}$ | ${X}_{21}{X}_{31}$ | ${X}_{21}{X}_{41}$ | ||||

${X}_{41}{X}_{31}$ | ${X}_{31}{X}_{41}$ |

Folded Code Sequences | Code Sequences | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

X_{62}X_{02} | ${X}_{52}{X}_{02}$ | ${X}_{42}{X}_{02}$ | ${X}_{32}{X}_{02}$ | ${X}_{22}{X}_{02}$ | ${X}_{12}{X}_{02}$ | ${X}_{02}$ | ${X}_{02}{X}_{22}$ | ${X}_{02}{X}_{32}$ | ${X}_{02}{X}_{42}$ | ${X}_{02}{X}_{52}$ | X_{02}X_{62} |

${X}_{62}{X}_{12}$ | ${X}_{52}{X}_{12}$ | ${X}_{42}{X}_{12}$ | ${X}_{32}{X}_{12}$ | ${X}_{22}{X}_{12}$ | ${X}_{12}{X}_{22}$ | ${X}_{12}{X}_{32}$ | ${X}_{12}{X}_{42}$ | ${X}_{12}{X}_{52}$ | ${X}_{12}{X}_{62}$. | ||

${X}_{62}{X}_{22}$. | ${X}_{52}{X}_{22}$ | ${X}_{42}{X}_{22}$ | ${X}_{12}{X}_{62}$ | ${X}_{22}{X}_{32}$ | ${X}_{22}{X}_{42}$ | ${X}_{22}{X}_{52}$ | ${X}_{22}{X}_{62}$ | ||||

${X}_{62}{X}_{32}$ | ${X}_{52}{X}_{32}$ | ${X}_{42}{X}_{32}$ | ${X}_{32}{X}_{42}$ | ${X}_{32}{X}_{52}$ | ${X}_{32}{X}_{62}$ | ||||||

${X}_{62}{X}_{42}$ | ${X}_{52}{X}_{42}$ | ${X}_{42}{X}_{52}$ | ${X}_{42}{X}_{62}$ | ||||||||

${X}_{62}{X}_{52}$ | ${X}_{52}{X}_{62}$ |

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## Share and Cite

**MDPI and ACS Style**

Morsy, M.A.; Aly, M.H.
A New Hybrid Prime Code for OCDMA Network Multimedia Applications. *Electronics* **2021**, *10*, 2705.
https://doi.org/10.3390/electronics10212705

**AMA Style**

Morsy MA, Aly MH.
A New Hybrid Prime Code for OCDMA Network Multimedia Applications. *Electronics*. 2021; 10(21):2705.
https://doi.org/10.3390/electronics10212705

**Chicago/Turabian Style**

Morsy, Morsy A., and Moustafa H. Aly.
2021. "A New Hybrid Prime Code for OCDMA Network Multimedia Applications" *Electronics* 10, no. 21: 2705.
https://doi.org/10.3390/electronics10212705